Normal subgroup

<h2 id="definitions">Definitions</h2>
A <a href="/facts/Subgroup/r91mBgpa">subgroup</a> 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of a group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is called a normal subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 if it is invariant under <a href="/facts/Inner_automorphism/awJPzhDa">conjugation</a>; that is, the conjugation of an element of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 by an element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is always in 
 
 
 
 N
 .
 
 
 {\displaystyle N.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The usual notation for this relation is 
 
 
 
 N
 ◃
 G
 .
 
 
 {\displaystyle N\triangleleft G.}

<h3>Equivalent conditions</h3>
For any subgroup 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 the following conditions are <a href="/facts/Logical_equivalence/ysCWmsxA">equivalent</a> to 
 
 
 
 N
 
 
 {\displaystyle N}
 
 being a normal subgroup of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 Therefore, any one of them may be taken as the definition.

<ul><li>The image of conjugation of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 by any element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a subset of 
 
 
 
 N
 ,
 
 
 {\displaystyle N,}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> i.e., 
 
 
 
 g
 N
 
 g
 
 −
 1
 
 
 ⊆
 N
 
 
 {\displaystyle gNg^{-1}\subseteq N}
 
 for all 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
.</li>
<li>The image of conjugation of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 by any element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is equal to 
 
 
 
 N
 ,
 
 
 {\displaystyle N,}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> i.e., 
 
 
 
 g
 N
 
 g
 
 −
 1
 
 
 =
 N
 
 
 {\displaystyle gNg^{-1}=N}
 
 for all 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
.</li>
<li>For all 
 
 
 
 g
 ∈
 G
 ,
 
 
 {\displaystyle g\in G,}
 
 the left and right cosets 
 
 
 
 g
 N
 
 
 {\displaystyle gN}
 
 and 
 
 
 
 N
 g
 
 
 {\displaystyle Ng}
 
 are equal.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>The sets of left and right <a href="/facts/Coset/zFgQUTLY">cosets</a> of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 coincide.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>Multiplication in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 preserves the equivalence relation "is in the same left coset as". That is, for every 
 
 
 
 g
 ,
 
 g
 ′
 
 ,
 h
 ,
 
 h
 ′
 
 ∈
 G
 
 
 {\displaystyle g,g',h,h'\in G}
 
 satisfying 
 
 
 
 g
 N
 =
 
 g
 ′
 
 N
 
 
 {\displaystyle gN=g'N}
 
 and 
 
 
 
 h
 N
 =
 
 h
 ′
 
 N
 
 
 {\displaystyle hN=h'N}
 
, we have 
 
 
 
 (
 g
 h
 )
 N
 =
 (
 
 g
 ′
 
 
 h
 ′
 
 )
 N
 .
 
 
 {\displaystyle (gh)N=(g'h')N.}
 
</li>
<li>There exists a group on the set of left cosets of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 where multiplication of any two left cosets 
 
 
 
 g
 N
 
 
 {\displaystyle gN}
 
 and 
 
 
 
 h
 N
 
 
 {\displaystyle hN}
 
 yields the left coset 
 
 
 
 (
 g
 h
 )
 N
 
 
 {\displaystyle (gh)N}
 
. (This group is called the quotient group of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 modulo 
 
 
 
 N
 
 
 {\displaystyle N}
 
, denoted 
 
 
 
 G
 
 /
 
 N
 
 
 {\displaystyle G/N}
 
.)</li>
<li>
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of <a href="/facts/Conjugacy_class/AFuPIFNW">conjugacy classes</a> of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>
 
 
 
 N
 
 
 {\displaystyle N}
 
 is preserved by the <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphisms</a> of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>There is some <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> 
 
 
 
 G
 →
 H
 
 
 {\displaystyle G\to H}
 
 whose <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> is 
 
 
 
 N
 .
 
 
 {\displaystyle N.}
 
<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>There exists a group homomorphism 
 
 
 
 ϕ
 :
 G
 →
 H
 
 
 {\displaystyle \phi :G\to H}
 
 whose <a href="/facts/Fiber_(mathematics)/rL3pwFwa">fibers</a> form a group where the identity element is 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and multiplication of any two fibers 
 
 
 
 
 ϕ
 
 −
 1
 
 
 (
 
 h
 
 1
 
 
 )
 
 
 {\displaystyle \phi ^{-1}(h_{1})}
 
 and 
 
 
 
 
 ϕ
 
 −
 1
 
 
 (
 
 h
 
 2
 
 
 )
 
 
 {\displaystyle \phi ^{-1}(h_{2})}
 
 yields the fiber 
 
 
 
 
 ϕ
 
 −
 1
 
 
 (
 
 h
 
 1
 
 
 
 h
 
 2
 
 
 )
 
 
 {\displaystyle \phi ^{-1}(h_{1}h_{2})}
 
. (This group is the same group 
 
 
 
 G
 
 /
 
 N
 
 
 {\displaystyle G/N}
 
 mentioned above.)</li>
<li>There is some <a href="/facts/Congruence_relation/TV3njpqF">congruence relation</a> on 
 
 
 
 G
 
 
 {\displaystyle G}
 
 for which the <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> of the <a href="/facts/Identity_element/9nss3xHY">identity element</a> is 
 
 
 
 N
 
 
 {\displaystyle N}
 
.</li>
<li>For all 
 
 
 
 n
 ∈
 N
 
 
 {\displaystyle n\in N}
 
 and 
 
 
 
 g
 ∈
 G
 ,
 
 
 {\displaystyle g\in G,}
 
 the <a href="/facts/Commutator/eYxzFmFY">commutator</a> 
 
 
 
 [
 n
 ,
 g
 ]
 =
 
 n
 
 −
 1
 
 
 
 g
 
 −
 1
 
 
 n
 g
 
 
 {\displaystyle [n,g]=n^{-1}g^{-1}ng}
 
 is in 
 
 
 
 N
 .
 
 
 {\displaystyle N.}
 
</li>
<li>Any two elements commute modulo the normal subgroup membership relation. That is, for all 
 
 
 
 g
 ,
 h
 ∈
 G
 ,
 
 
 {\displaystyle g,h\in G,}
 
 
 
 
 
 g
 h
 ∈
 N
 
 
 {\displaystyle gh\in N}
 
 if and only if 
 
 
 
 h
 g
 ∈
 N
 .
 
 
 {\displaystyle hg\in N.}
 
</li></ul>
<h2 id="examples">Examples</h2>
For any group 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 the trivial subgroup 
 
 
 
 {
 e
 }
 
 
 {\displaystyle \{e\}}
 
 consisting of just the identity element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is always a normal subgroup of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 Likewise, 
 
 
 
 G
 
 
 {\displaystyle G}
 
 itself is always a normal subgroup of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 (If these are the only normal subgroups, then 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is said to be <a href="/facts/Simple_group/e5hGF7nT">simple</a>.)<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Other named normal subgroups of an arbitrary group include the <a href="/facts/Center_(group_theory)/gqAg6B8I">center of the group</a> (the set of elements that commute with all other elements) and the <a href="/facts/Commutator_subgroup/oK9WphdE">commutator subgroup</a> 
 
 
 
 [
 G
 ,
 G
 ]
 .
 
 
 {\displaystyle [G,G].}
 
<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> More generally, since conjugation is an isomorphism, any <a href="/facts/Characteristic_subgroup/UfoS7U7S">characteristic subgroup</a> is a normal subgroup.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> then every subgroup 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is normal, because 
 
 
 
 g
 N
 =
 {
 g
 n
 
 }
 
 n
 ∈
 N
 
 
 =
 {
 n
 g
 
 }
 
 n
 ∈
 N
 
 
 =
 N
 g
 .
 
 
 {\displaystyle gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.}
 
 More generally, for any group 
 
 
 
 G
 
 
 {\displaystyle G}
 
, every subgroup of the center 
 
 
 
 Z
 (
 G
 )
 
 
 {\displaystyle Z(G)}
 
 of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is normal in 
 
 
 
 G
 
 
 {\displaystyle G}
 
. (In the special case that 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is abelian, the center is all of 
 
 
 
 G
 
 
 {\displaystyle G}
 
, hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a <a href="/facts/Hamiltonian_group/3xQ7BNxe">Hamiltonian group</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
A concrete example of a normal subgroup is the subgroup 
 
 
 
 N
 =
 {
 (
 1
 )
 ,
 (
 123
 )
 ,
 (
 132
 )
 }
 
 
 {\displaystyle N=\{(1),(123),(132)\}}
 
 of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> 
 
 
 
 
 S
 
 3
 
 
 ,
 
 
 {\displaystyle S_{3},}
 
 consisting of the identity and both three-cycles. In particular, one can check that every coset of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is either equal to 
 
 
 
 N
 
 
 {\displaystyle N}
 
 itself or is equal to 
 
 
 
 (
 12
 )
 N
 =
 {
 (
 12
 )
 ,
 (
 23
 )
 ,
 (
 13
 )
 }
 .
 
 
 {\displaystyle (12)N=\{(12),(23),(13)\}.}
 
 On the other hand, the subgroup 
 
 
 
 H
 =
 {
 (
 1
 )
 ,
 (
 12
 )
 }
 
 
 {\displaystyle H=\{(1),(12)\}}
 
 is not normal in 
 
 
 
 
 S
 
 3
 
 
 
 
 {\displaystyle S_{3}}
 
 since 
 
 
 
 (
 123
 )
 H
 =
 {
 (
 123
 )
 ,
 (
 13
 )
 }
 ≠
 {
 (
 123
 )
 ,
 (
 23
 )
 }
 =
 H
 (
 123
 )
 .
 
 
 {\displaystyle (123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).}
 
<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> This illustrates the general fact that any subgroup 
 
 
 
 H
 ≤
 G
 
 
 {\displaystyle H\leq G}
 
 of index two is normal.
As an example of a normal subgroup within a <a href="/facts/Matrix_group/wVzG0sa3">matrix group</a>, consider the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> 
 
 
 
 
 
 G
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )}
 
 of all invertible 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrices with real entries under the operation of matrix multiplication and its subgroup 
 
 
 
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}
 
 of all 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrices of <a href="/facts/Determinant_(mathematics)/eGYqJ2TF">determinant</a> 1 (the <a href="/facts/Special_linear_group/rsahah9w">special linear group</a>). To see why the subgroup 
 
 
 
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}
 
 is normal in 
 
 
 
 
 
 G
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )}
 
, consider any matrix 
 
 
 
 X
 
 
 {\displaystyle X}
 
 in 
 
 
 
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}
 
 and any invertible matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
. Then using the two important identities 
 
 
 
 det
 (
 A
 B
 )
 =
 det
 (
 A
 )
 det
 (
 B
 )
 
 
 {\displaystyle \det(AB)=\det(A)\det(B)}
 
 and 
 
 
 
 det
 (
 
 A
 
 −
 1
 
 
 )
 =
 det
 (
 A
 
 )
 
 −
 1
 
 
 
 
 {\displaystyle \det(A^{-1})=\det(A)^{-1}}
 
, one has that 
 
 
 
 det
 (
 A
 X
 
 A
 
 −
 1
 
 
 )
 =
 det
 (
 A
 )
 det
 (
 X
 )
 det
 (
 A
 
 )
 
 −
 1
 
 
 =
 det
 (
 X
 )
 =
 1
 
 
 {\displaystyle \det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1}
 
, and so 
 
 
 
 A
 X
 
 A
 
 −
 1
 
 
 ∈
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )}
 
 as well. This means 
 
 
 
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}
 
 is closed under conjugation in 
 
 
 
 
 
 G
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )}
 
, so it is a normal subgroup.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
In the <a href="/facts/Rubik%2527s_Cube_group/pFsPzCKU">Rubik's Cube group</a>, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
The <a href="/facts/Translation_group/KlpWmnfX">translation group</a> is a normal subgroup of the <a href="/facts/Euclidean_group/VdTSdwzl">Euclidean group</a> in any dimension.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all <a href="/facts/Rotation/yChkz49x">rotations</a> about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.

<h2 id="properties">Properties</h2>
<ul><li>If 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a normal subgroup of 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 and 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 containing 
 
 
 
 H
 ,
 
 
 {\displaystyle H,}
 
 then 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a normal subgroup of 
 
 
 
 K
 .
 
 
 {\displaystyle K.}
 
<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li>
<li>A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a <a href="/facts/Transitive_relation/9r90y50r">transitive relation</a>. The smallest group exhibiting this phenomenon is the <a href="/facts/Dihedral_group/1PUlRh4M">dihedral group</a> of order 8.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> However, a <a href="/facts/Characteristic_subgroup/UfoS7U7S">characteristic subgroup</a> of a normal subgroup is normal.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> A group in which normality is transitive is called a <a href="/facts/T-group_(mathematics)/285Gkuvq">T-group</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>The two groups 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and 
 
 
 
 H
 
 
 {\displaystyle H}
 
 are normal subgroups of their <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct product</a> 
 
 
 
 G
 ×
 H
 .
 
 
 {\displaystyle G\times H.}
 
</li>
<li>If the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Semidirect_product/Uqf87k1I">semidirect product</a> 
 
 
 
 G
 =
 N
 ⋊
 H
 ,
 
 
 {\displaystyle G=N\rtimes H,}
 
 then 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is normal in 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 though 
 
 
 
 H
 
 
 {\displaystyle H}
 
 need not be normal in 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
</li>
<li>If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are normal subgroups of an additive group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 such that 
 
 
 
 G
 =
 M
 +
 N
 
 
 {\displaystyle G=M+N}
 
 and 
 
 
 
 M
 ∩
 N
 =
 {
 0
 }
 
 
 {\displaystyle M\cap N=\{0\}}
 
, then 
 
 
 
 G
 =
 M
 ⊕
 N
 .
 
 
 {\displaystyle G=M\oplus N.}
 
<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li>Normality is preserved under surjective homomorphisms;<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> that is, if 
 
 
 
 G
 →
 H
 
 
 {\displaystyle G\to H}
 
 is a surjective group homomorphism and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is normal in 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 then the image 
 
 
 
 f
 (
 N
 )
 
 
 {\displaystyle f(N)}
 
 is normal in 
 
 
 
 H
 .
 
 
 {\displaystyle H.}
 
</li>
<li>Normality is preserved by taking <a href="/facts/Inverse_image/KANefMAl">inverse images</a>;<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> that is, if 
 
 
 
 G
 →
 H
 
 
 {\displaystyle G\to H}
 
 is a group homomorphism and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is normal in 
 
 
 
 H
 ,
 
 
 {\displaystyle H,}
 
 then the inverse image 
 
 
 
 
 f
 
 −
 1
 
 
 (
 N
 )
 
 
 {\displaystyle f^{-1}(N)}
 
 is normal in 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
</li>
<li>Normality is preserved on taking <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct products</a>;<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> that is, if 
 
 
 
 
 N
 
 1
 
 
 ◃
 
 G
 
 1
 
 
 
 
 {\displaystyle N_{1}\triangleleft G_{1}}
 
 and 
 
 
 
 
 N
 
 2
 
 
 ◃
 
 G
 
 2
 
 
 ,
 
 
 {\displaystyle N_{2}\triangleleft G_{2},}
 
 then 
 
 
 
 
 N
 
 1
 
 
 ×
 
 N
 
 2
 
 
 
 ◃
 
 
 G
 
 1
 
 
 ×
 
 G
 
 2
 
 
 .
 
 
 {\displaystyle N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}.}
 
</li>
<li>Every subgroup of <a href="/facts/Index_(group_theory)/ewY2A71Y">index</a> 2 is normal. More generally, a subgroup, 
 
 
 
 H
 ,
 
 
 {\displaystyle H,}
 
 of finite index, 
 
 
 
 n
 ,
 
 
 {\displaystyle n,}
 
 in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 contains a subgroup, 
 
 
 
 K
 ,
 
 
 {\displaystyle K,}
 
 normal in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and of index dividing 
 
 
 
 n
 !
 
 
 {\displaystyle n!}
 
 called the <a href="/facts/Normal_core/IGEJE92q">normal core</a>. In particular, if 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is the smallest prime dividing the order of 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 then every subgroup of index 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is normal.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a></li>
<li>The fact that normal subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 are precisely the kernels of group homomorphisms defined on 
 
 
 
 G
 
 
 {\displaystyle G}
 
 accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is <a href="/facts/Simple_group/e5hGF7nT">simple</a> if and only if it is isomorphic to all of its non-identity homomorphic images,<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> a finite group is <a href="/facts/Perfect_group/rxztKKjd">perfect</a> if and only if it has no normal subgroups of prime <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a>, and a group is <a href="/facts/Imperfect_group/yne7cJuj">imperfect</a> if and only if the <a href="/facts/Derived_subgroup/oK9WphdE">derived subgroup</a> is not supplemented by any proper normal subgroup.</li></ul>
<h3>Lattice of normal subgroups</h3>
Given two normal subgroups, 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and 
 
 
 
 M
 ,
 
 
 {\displaystyle M,}
 
 of 
 
 
 
 G
 ,
 
 
 {\displaystyle G,}
 
 their intersection 
 
 
 
 N
 ∩
 M
 
 
 {\displaystyle N\cap M}
 
and their <a href="/facts/Product_of_subgroups/I8Agzkqr">product</a> 
 
 
 
 N
 M
 =
 {
 n
 m
 :
 n
 ∈
 N
 
 
  and 
 
 
 m
 ∈
 M
 }
 
 
 {\displaystyle NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}}
 
 are also normal subgroups of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 
The normal subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 form a <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> under <a href="/facts/Subset_inclusion/SJGERmbA">subset inclusion</a> with <a href="/facts/Least_element/JVn2B3mo">least element</a>, 
 
 
 
 {
 e
 }
 ,
 
 
 {\displaystyle \{e\},}
 
 and <a href="/facts/Greatest_element/JVn2B3mo">greatest element</a>, 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 The <a href="/facts/Meet_(lattice_theory)/5ak6ufky">meet</a> of two normal subgroups, 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and 
 
 
 
 M
 ,
 
 
 {\displaystyle M,}
 
 in this lattice is their intersection and the <a href="/facts/Join_(lattice_theory)/5ak6ufky">join</a> is their product.
The lattice is <a href="/facts/Complete_lattice/pQbphWeR">complete</a> and <a href="/facts/Modular_lattice/CLe2aDq7">modular</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

<h2 id="normal-subgroups-quotient-groups-and-homomorphisms">Normal subgroups, quotient groups and homomorphisms</h2>
If 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a normal subgroup, we can define a multiplication on cosets as follows:

(
          
            
              a
              
                1
              
            
            N
          
          )
        
        
          (
          
            
              a
              
                2
              
            
            N
          
          )
        
        :=
        
          (
          
            
              a
              
                1
              
            
            
              a
              
                2
              
            
          
          )
        
        N
        .
      
    
    {\displaystyle \left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N.}

This relation defines a mapping 
 
 
 
 G
 
 /
 
 N
 ×
 G
 
 /
 
 N
 →
 G
 
 /
 
 N
 .
 
 
 {\displaystyle G/N\times G/N\to G/N.}
 
 To show that this mapping is well-defined, one needs to prove that the choice of representative elements 
 
 
 
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 
 
 {\displaystyle a_{1},a_{2}}
 
 does not affect the result. To this end, consider some other representative elements 
 
 
 
 
 a
 
 1
 
 ′
 
 ∈
 
 a
 
 1
 
 
 N
 ,
 
 a
 
 2
 
 ′
 
 ∈
 
 a
 
 2
 
 
 N
 .
 
 
 {\displaystyle a_{1}'\in a_{1}N,a_{2}'\in a_{2}N.}
 
 Then there are 
 
 
 
 
 n
 
 1
 
 
 ,
 
 n
 
 2
 
 
 ∈
 N
 
 
 {\displaystyle n_{1},n_{2}\in N}
 
 such that 
 
 
 
 
 a
 
 1
 
 ′
 
 =
 
 a
 
 1
 
 
 
 n
 
 1
 
 
 ,
 
 a
 
 2
 
 ′
 
 =
 
 a
 
 2
 
 
 
 n
 
 2
 
 
 .
 
 
 {\displaystyle a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}.}
 
 It follows that 
 
 
 
 
 a
 
 1
 
 ′
 
 
 a
 
 2
 
 ′
 
 N
 =
 
 a
 
 1
 
 
 
 n
 
 1
 
 
 
 a
 
 2
 
 
 
 n
 
 2
 
 
 N
 =
 
 a
 
 1
 
 
 
 a
 
 2
 
 
 
 n
 
 1
 
 ′
 
 
 n
 
 2
 
 
 N
 =
 
 a
 
 1
 
 
 
 a
 
 2
 
 
 N
 ,
 
 
 {\displaystyle a_{1}'a_{2}'N=a_{1}n_{1}a_{2}n_{2}N=a_{1}a_{2}n_{1}'n_{2}N=a_{1}a_{2}N,}
 
where we also used the fact that 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a normal subgroup, and therefore there is 
 
 
 
 
 n
 
 1
 
 ′
 
 ∈
 N
 
 
 {\displaystyle n_{1}'\in N}
 
 such that 
 
 
 
 
 n
 
 1
 
 
 
 a
 
 2
 
 
 =
 
 a
 
 2
 
 
 
 n
 
 1
 
 ′
 
 .
 
 
 {\displaystyle n_{1}a_{2}=a_{2}n_{1}'.}
 
 This proves that this product is a well-defined mapping between cosets.
With this operation, the set of cosets is itself a group, called the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> and denoted with 
 
 
 
 G
 
 /
 
 N
 .
 
 
 {\displaystyle G/N.}
 
 There is a natural <a href="/facts/Group_homomorphism/sDguaNYs">homomorphism</a>, 
 
 
 
 f
 :
 G
 →
 G
 
 /
 
 N
 ,
 
 
 {\displaystyle f:G\to G/N,}
 
 given by 
 
 
 
 f
 (
 a
 )
 =
 a
 N
 .
 
 
 {\displaystyle f(a)=aN.}
 
 This homomorphism maps 
 
 
 
 N
 
 
 {\displaystyle N}
 
 into the identity element of 
 
 
 
 G
 
 /
 
 N
 ,
 
 
 {\displaystyle G/N,}
 
 which is the coset 
 
 
 
 e
 N
 =
 N
 ,
 
 
 {\displaystyle eN=N,}
 
<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> that is, 
 
 
 
 ker
 ⁡
 (
 f
 )
 =
 N
 .
 
 
 {\displaystyle \ker(f)=N.}

In general, a group homomorphism, 
 
 
 
 f
 :
 G
 →
 H
 
 
 {\displaystyle f:G\to H}
 
 sends subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 to subgroups of 
 
 
 
 H
 .
 
 
 {\displaystyle H.}
 
 Also, the preimage of any subgroup of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a subgroup of 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
 We call the preimage of the trivial group 
 
 
 
 {
 e
 }
 
 
 {\displaystyle \{e\}}
 
 in 
 
 
 
 H
 
 
 {\displaystyle H}
 
 the <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> of the homomorphism and denote it by 
 
 
 
 ker
 ⁡
 f
 .
 
 
 {\displaystyle \ker f.}
 
 As it turns out, the kernel is always normal and the image of 
 
 
 
 G
 ,
 f
 (
 G
 )
 ,
 
 
 {\displaystyle G,f(G),}
 
 is always <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to 
 
 
 
 G
 
 /
 
 ker
 ⁡
 f
 
 
 {\displaystyle G/\ker f}
 
 (the <a href="/facts/First_isomorphism_theorem/SNpPTkrd">first isomorphism theorem</a>).<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> In fact, this correspondence is a bijection between the set of all quotient groups of 
 
 
 
 G
 ,
 G
 
 /
 
 N
 ,
 
 
 {\displaystyle G,G/N,}
 
 and the set of all homomorphic images of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 (<a href="/facts/Up_to/ydz4ztzX">up to</a> isomorphism).<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> It is also easy to see that the kernel of the quotient map, 
 
 
 
 f
 :
 G
 →
 G
 
 /
 
 N
 ,
 
 
 {\displaystyle f:G\to G/N,}
 
 is 
 
 
 
 N
 
 
 {\displaystyle N}
 
 itself, so the normal subgroups are precisely the kernels of homomorphisms with <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
 
 
 
 G
 .
 
 
 {\displaystyle G.}
 
<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<h2 id="see-also">See also</h2>

<h3>Operations taking subgroups to subgroups</h3>
<ul><li><a href="/facts/Normalizer/AKhrEUGR">Normalizer</a></li>
<li><a href="/facts/Normal_closure_(group_theory)/k2lJahCW">Normal closure</a></li>
<li><a href="/facts/Normal_core/IGEJE92q">Normal core</a></li></ul>
<h3>Subgroup properties complementary (or opposite) to normality</h3>
<ul><li><a href="/facts/Malnormal_subgroup/BFG1Lgqp">Malnormal subgroup</a></li>
<li><a href="/facts/Contranormal_subgroup/MSudXsJ1">Contranormal subgroup</a></li>
<li><a href="/facts/Abnormal_subgroup/B5RsfRUy">Abnormal subgroup</a></li>
<li><a href="/facts/Self-normalizing_subgroup/AKhrEUGR">Self-normalizing subgroup</a></li></ul>
<h3>Subgroup properties stronger than normality</h3>
<ul><li><a href="/facts/Characteristic_subgroup/UfoS7U7S">Characteristic subgroup</a></li>
<li><a href="/facts/Fully_characteristic_subgroup/UfoS7U7S">Fully characteristic subgroup</a></li></ul>
<h3>Subgroup properties weaker than normality</h3>
<ul><li><a href="/facts/Subnormal_subgroup/db2Koonc">Subnormal subgroup</a></li>
<li><a href="/facts/Ascendant_subgroup/ee1ol9Tr">Ascendant subgroup</a></li>
<li><a href="/facts/Descendant_subgroup/YbbNPo49">Descendant subgroup</a></li>
<li><a href="/facts/Quasinormal_subgroup/twYaQCf0">Quasinormal subgroup</a></li>
<li><a href="/facts/Seminormal_subgroup/GRVkGCIu">Seminormal subgroup</a></li>
<li><a href="/facts/Conjugate_permutable_subgroup/9P5EuMMU">Conjugate permutable subgroup</a></li>
<li><a href="/facts/Modular_subgroup/7DrwD6dc">Modular subgroup</a></li>
<li><a href="/facts/Pronormal_subgroup/bbF3Fbod">Pronormal subgroup</a></li>
<li><a href="/facts/Paranormal_subgroup/L21XFSG2">Paranormal subgroup</a></li>
<li><a href="/facts/Polynormal_subgroup/YqCJQqhj">Polynormal subgroup</a></li>
<li><a href="/facts/C-normal_subgroup/UIy4NImH">C-normal subgroup</a></li></ul>
<h3>Related notions in algebra</h3>
<ul><li><a href="/facts/Ideal_(ring_theory)/vCvytduX">Ideal (ring theory)</a></li>
<li><a href="/facts/Semigroup_ideal/rgSdk2kt">Semigroup ideal</a></li></ul>

<h2 id="notes">Notes</h2>

<h2 id="bibliography">Bibliography</h2>

<ul><li>Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). <a href="https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf">"On Rubik's Cube"</a> (PDF). <a href="/facts/KTH_Royal_Institute_of_Technology/VM9jmsXQ">KTH</a>.</li>
<li>Cantrell, C.D. (2000). <a href="https://archive.org/details/modernmathematic0000cant">Modern Mathematical Methods for Physicists and Engineers</a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-59180-5.</li>
<li>Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM.</li>
<li>Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-43334-9.</li>
<li>Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-321-15608-2.</li>
<li>Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1967-8.</li>
<li>Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer.</li>
<li>Hungerford, Thomas (2013). Abstract Algebra: An Introduction. Brooks/Cole Cengage Learning.</li>
<li>Judson, Thomas W. (2020). <a href="http://abstract.ups.edu/aata/aata.html">Abstract Algebra: Theory and Applications</a>.</li>
<li>Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-6443-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0836.20001">0836.20001</a>.</li>
<li><a href="/facts/William_Thurston/K1Qe8801">Thurston, William</a> (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08304-9.</li>
<li>Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-958258-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/859155300">859155300</a>.</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/I._N._Herstein/fWts4GeY">I. N. Herstein</a>, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/NormalSubgroup.html">"normal subgroup"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="https://encyclopediaofmath.org/wiki/Normal_subgroup">Normal subgroup in Springer's Encyclopedia of Mathematics</a></li>
<li><a href="http://www.math.uiuc.edu/~r-ash/Algebra/Chapter1.pdf">Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year</a></li>
<li><a href="http://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups">Timothy Gowers, Normal subgroups and quotient groups</a></li>
<li><a href="http://math.ucr.edu/home/baez/normal.html">John Baez, What's a Normal Subgroup?</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bradley 2010, p. 12. - Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 978-0-19-958258-7. OCLC 859155300. <a href="https://search.worldcat.org/oclc/859155300" target="_blank">https://search.worldcat.org/oclc/859155300</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cantrell 2000, p. 160. - Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5. <a href="https://archive.org/details/modernmathematic0000cant" target="_blank">https://archive.org/details/modernmathematic0000cant</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dummit & Foote 2004. - Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hungerford 2003, p. 41. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hungerford 2003, p. 41. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Hungerford 2003, p. 41. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hungerford 2003, p. 41. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Cantrell 2000, p. 160. - Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5. <a href="https://archive.org/details/modernmathematic0000cant" target="_blank">https://archive.org/details/modernmathematic0000cant</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Fraleigh 2003, p. 141. - Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Cantrell 2000, p. 160. - Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5. <a href="https://archive.org/details/modernmathematic0000cant" target="_blank">https://archive.org/details/modernmathematic0000cant</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Robinson 1996, p. 16. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Hungerford 2003, p. 45. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Hall 1999, p. 138. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Hall 1999, p. 32. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Hall 1999, p. 190. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Judson 2020, Section 10.1. - Judson, Thomas W. (2020). Abstract Algebra: Theory and Applications. <a href="http://abstract.ups.edu/aata/aata.html" target="_blank">http://abstract.ups.edu/aata/aata.html</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">In other language: 
 
 
 
 det
 
 
 {\displaystyle \det }
 
 is a homomorphism from 
 
 
 
 
 
 G
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )}
 
 to the multiplicative subgroup 
 
 
 
 
 
 R
 
 
 ×
 
 
 
 
 {\displaystyle \mathbf {R} ^{\times }}
 
, and 
 
 
 
 
 
 S
 L
 
 
 n
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}
 
 is the kernel. Both arguments also work over the complex numbers, or indeed over an arbitrary field. <a href="/wiki/Complex_number" target="_blank">/wiki/Complex_number</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Bergvall et al. 2010, p. 96. - Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. <a href="https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf" target="_blank">https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Thurston 1997, p. 218. - Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Hungerford 2003, p. 42. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Robinson 1996, p. 17. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Robinson 1996, p. 28. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Robinson 1996, p. 402. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Hungerford 2013, p. 290. - Hungerford, Thomas (2013). Abstract Algebra: An Introduction. Brooks/Cole Cengage Learning. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Hall 1999, p. 29. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Hall 1999, p. 29. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Hungerford 2003, p. 46. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Robinson 1996, p. 36. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Dõmõsi & Nehaniv 2004, p. 7. - Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Hungerford 2003, p. 46. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Hungerford 2003, pp. 42–43. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Hungerford 2003, p. 44. - Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Robinson 1996, p. 20. - Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. <a href="https://zbmath.org/?format=complete&q=an:0836.20001" target="_blank">https://zbmath.org/?format=complete&q=an:0836.20001</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Hall 1999, p. 27. - Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
</ol>

Normal subgroup open-in-new

Normal subgroup