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The celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence 1 -> N -> (iota) G -> (pi) H -> 1 of groups and a section s:H -> G, an embedding Phi : G -> N(sic)H of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by (G;iota,pi). Moreover, one says that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) of H by N are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that iota(2)=eta circle iota(1) and pi(1)=pi(2)circle eta. We say that two embeddings Phi(1):G(1) -> N(sic)H and Phi(2):G(2)-> N(sic)H are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that Phi(1)=Phi(2) circle eta. We show that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) are equivalent if and only if the embeddings Phi(1) and Phi(2), associated with any two sections s(1 ): H -> G(1) and s(2 ): H -> G(2) via the Kaloujnine-Krasner theorem, are equivalent.

© 2016, Copyright © Taylor & Francis Group, LLC. We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra 𝔤, an ideal 𝔥 of 𝔤, and the quotient 𝔤/𝔥 with coefficients in a 𝔤-module. The existence of such a sequence follows from the Hochschild–Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions, etc. The maps are constructed using elementary methods. This alternative approach to the seven term exact sequence can certainly be useful, especially since we include straightforward cocycle descriptions of the constructed maps. peerreview_statement: The publishing and review policy for this title is described in its Aims & Scope. aims_and_scope_url: http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=lagb20 ispartof: Communications in Algebra vol:44 issue:3 pages:1321-1349 status: published

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that the FC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence the FC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group.

Let G be a finite group with a normal subgroup N that has a p-block b of defect 0. As is well-known, we may assume that b is G-stable. Then we show that there is an extension of G/N by a central cyclic p′-subgroup and a p-block of such that the blocks of G covering b biject with the blocks of that cover and corresponding blocks have several equivalent properties. These results extend the known results in the situation that N is a normal p′-subgroup.

For a certain class of groups, which are semidirect products arising from an action of a finite rank free abelian group on another group, we study cancellation of the infinite cyclic group in isomorphic direct products. As an application we obtain a sufficient condition for triviality of the genus of certain nilpotent groups.

The construction used in Schur's central extension theorem is generalized by proving that given a cocommutative Hopf algebra H, there is a cocornmutative central extension B of H such that any projective representation of H lifts to an ordinary representation of B. The extension B is the crossed product of H with the finite dual of the group algebra kZ1, where Z1denotes the group of normal cocycles on H with values in the ground field k.

Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σ n , where σ n is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel...

In this paper, we find an exact sequence involving the automorphism group of a self-dual type-A complex lattice (including the Niemeier lattice A12 2 and the complex Leech lattice) and the automorphism of a certain group central extension of the lattice. Such an exact sequence is important for the study of finite groups which have a moonshine representation analogous to that of the Monster

A finite group G is called Q-admissible if there exists a finite-dimensional central division algebra over Q which is a crossed product for G.It is proved that every Sylow-metacyclic group having .as quotient group is Q-admissible, where is tne extension of S5 by the group of order 2 with the generalized quaternion group of order 16 as a Sylow 2-subgroup.