Capability, Homology, and Central Series of a Pair of Groups

The Exterior Degree of a Pair of Finite Groups

“…The exterior product G ∧ N is obtained from G ⊗ N by imposing the additional relation x ⊗ x for all x in N and the image of g ⊗ n is denoted by g ∧ n for all g ∈ G, n ∈ N . By using the notations in [4], the exterior G-centre of N is a central subgroup of N which is defined as follows…”

“…Let G = C 4 and N = 2C4 . Then[4, Corollary 8] shows that the pair (C 4 , 2C 4 ) is capable, and sod ∧ (C 4 , 2C 4 ) = 1 |2C 4 ||C 4 | x∈C4 | Let G = D 8and N be the subgroup a 2 , ab of G. A computation similar to Example 4.1 shows that the pair (D 8 , a 2 , ab ) is capable, andd ∧ (D 8 , a 2 , ab ) = 1 | a 2 , ab ||D 8 | x∈D8 | a 2 ,ab C ∧ (…”

The precise center of a crossed module

“…This property will provide us with some computable expressions for Z Ã ðT; G; qÞ. In Section 6 we study connections between capable crossed modules and some other capability notions developed in [9] and [17], like capable pairs of groups or relatively capable groups.…”

“…! ðT; G; qÞ with i injective (see Example 12); then [9] the research on relatively capable groups, and proposed an extension of capability theory for groups to a theory for pairs of groups. By a pair of groups ðG; NÞ he understands a group G and a normal subgroup N. A capable pair is a pair of groups ðG; NÞ such that there exists a group M and a crossed module q : M !…”

The commutator subgroup and Schur multiplier of a pair of finite p-groups

“…Ali Reza Salemkar, Mohammad Reza R. Moghaddam and Farshid Saeedi [2] Now for the given relative central extension a, we define G-commutator and Gcentral subgroups of N, respectively, as follows In special case a = i, [M, G] and Z(M, G) are the commutator subgroup and the centralizer of G in M, respectively. In this case, we define…”