Regular p-group

<h2 id="definition">Definition</h2>
A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch. 12.4), (Huppert 1967, Kap. III §10) conditions are satisfied:

<ul><li>For every a, b in G, there is a c in the <a href="/facts/Derived_subgroup/oK9WphdE">derived subgroup</a> H′ of the subgroup H of G generated by a and b, such that ap · bp = (ab)p · cp.</li>
<li>For every a, b in G, there are elements ci in the derived subgroup of the subgroup generated by a and b, such that ap · bp = (ab)p · c1p ⋯ ckp.</li>
<li>For every a, b in G and every positive integer n, there are elements ci in the derived subgroup of the subgroup generated by a and b such that aq · bq = (ab)q · c1q ⋯ ckq, where q = pn.</li></ul>
<h2 id="examples">Examples</h2>
Many familiar p-groups are regular:

<ul><li>Every <a href="/facts/Abelian_group/FfWwmx4R">abelian</a> p-group is regular.</li>
<li>Every p-group of <a href="/facts/Nilpotency_class/BJuochky">nilpotency class</a> strictly less than p is regular. This follows from the <a href="/facts/Hall%25E2%2580%2593Petresco_identity/mLge2sUd">Hall–Petresco identity</a>.</li>
<li>Every p-group of <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> at most pp is regular.</li>
<li>Every finite group of <a href="/facts/Exponent_(group_theory)/TEvLhDtm">exponent</a> p is regular.</li></ul>
However, many familiar p-groups are not regular:

<ul><li>Every nonabelian 2-group is irregular.</li>
<li>The <a href="/facts/Sylow_subgroup/aVTxDRUo">Sylow p-subgroup</a> of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> on p2 points is irregular and of order pp+1.</li></ul>
<h2 id="properties">Properties</h2>
A p-group is regular <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> every <a href="/facts/Subgroup/r91mBgpa">subgroup</a> generated by two elements is regular.
Every subgroup and <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> of a regular group is regular, but the <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct product</a> of regular groups need not be regular.
A 2-group is regular if and only if it is abelian. A 3-group with two generators is regular if and only if its derived subgroup is <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>. Every p-group of odd order with cyclic derived subgroup is regular.
The subgroup of a p-group G generated by the elements of order dividing pk is denoted <a href="/facts/Omega_and_agemo_subgroup/C2gK9KwH">Ωk(G)</a> and regular groups are well-behaved in that Ωk(G) is precisely the set of elements of order dividing pk. The subgroup generated by all pk-th powers of elements in G is denoted <a href="/facts/Omega_and_agemo_subgroup/C2gK9KwH">℧k(G)</a>. In a regular group, the <a href="/facts/Index_(group_theory)/ewY2A71Y">index</a> [G:℧k(G)] is equal to the order of Ωk(G). In fact, commutators and powers interact in particularly simple ways (Huppert 1967, Kap III §10, Satz 10.8). For example, given normal subgroups M and N of a regular p-group G and nonnegative integers m and n, one has [℧m(M),℧n(N)] = ℧m+n([M,N]).

<ul><li><a href="/facts/Philip_Hall/4Gal202l">Philip Hall</a>'s criteria of regularity of a p-group G: G is regular, if one of the following hold:
<ol><li>[G:℧1(G)] < pp</li>
<li>[G′:℧1(G′)| < pp−1</li>
<li>|Ω1(G)| < pp−1</li></ol></li></ul>
<h2 id="generalizations">Generalizations</h2>
<ul><li><a href="/facts/Powerful_p-group/tDMgwvPa">Powerful p-group</a></li>
<li><a href="/facts/Power_closed/Qz4nRmeM">power closed</a> p-group</li></ul>

<ul><li><a href="/facts/Marshall_Hall_(mathematician)/wLpjsYnL">Hall, Marshall</a> (1959), The theory of groups, Macmillan, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0103215">0103215</a></li>
<li><a href="/facts/Philip_Hall/4Gal202l">Hall, Philip</a> (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-36.1.29">10.1112/plms/s2-36.1.29</a></li>
<li><a href="/facts/Bertram_Huppert/eEL3KWHQ">Huppert, B.</a> (1967), Endliche Gruppen (in German), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 90–93, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-03825-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0224703">0224703</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/527050">527050</a></li></ul>

Regular p-group open-in-new

Regular p-group