Regular p-group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Suppose p is a prime number. A p-group G (i.e., a group where the order of every element is a power of p) is termed a regular p-group if it satisfies the following equivalent conditions:

  1. For every a,b \in G, there exists c \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc^p.
  2. For every a,b \in G, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc_1^pc_2^p \dots c_k^p.
  3. For every a,b \in G and every natural number n, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^qb^q = (ab)^qc_1^qc_2^q \dots c_k^q where q = p^n.

The term regular p-group is typically used only for finite p-groups.

Relation with other properties

Stronger properties

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes regular p-group property is subgroup-closed If p is prime, G is a regular p-group, and H is a subgroup of G, then H is a regular p-group.
quotient-closed group property Yes regular p-group property is quotient-closed If p is prime, G is a regular p-group, H is a normal subgroup of G, and G/H is the corresponding quotient group, then G/H is also a regular p-group.
finite direct product-closed group property No regular p-group property is finite direct product-closed It is possible to have a prime number p and regular p-groups G_1,G_2 such that the external direct product G_1 \times G_2 is not a regular p-group.
2-local group property Yes regular p-group property is 2-local Suppose p is a prime number and G is a group such that for all a,b \in G, \langle a,b \rangle is a regular p-group, then G itself is a regular p-group.