A Note on Products of Normal Subgroups (original) (raw)
On normal subgroups of direct products
Proceedings of the Edinburgh Mathematical Society, 1990
We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.
A note on generalised wreath product groups
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1985
Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.
The HNN and generalized free product structure of certain linear groups
Bulletin of the American Mathematical Society, 1975
Introduction. If d is a positive square-free integer let I d be the ring of integers in Q (\J-d). I d is a Euclidean domain if d = 1, 2, 3, 7, 11. The groups PSL 2 (/ d ) = F d over these Euclidean rings have recently been investigated. Methods for generating presentations as well as actual presentations were given in [2], [3] and [8], while these groups were shown to be describable in terms of generalized free products in . Here we announce several extensions of these results suggested by Karrass and Solitar. We show that the Picard group T x is decomposable directly as a free product with amalgamated subgroup while the groups r 2 , T 7 , T x t are HNN groups in the sense of . The extensions will be used in [4] to show that these groups are SQ-universal.
Journal of Group Theory, 2007
A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.
Bulletin of the Australian Mathematical Society, 1982
We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite...
Acta Mathematica Academiae Scientiarum Hungaricae
w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG
2015
In this paper we give some general results on the non-split extension group Gn = 2 2n· Sp(2n, 2), n ≥ 2. We then focus on the group G 4 = 2 8· Sp(8, 2). We construct G 4 as a permutation group acting on 512 points. The conjugacy classes are determined using the coset analysis technique. Then we determine the inertia factor groups and Fischer matrices, which are required for the computations of the character table of G 4 by means of Clifford-Fischer Theory. There are two inertia factor groups namely H 1 = Sp(8, 2) and H 2 = 2 7 :Sp(6, 2), the Schur multiplier and hence the character table of the corresponding covering group of H 2 were calculated. Using the information on conjugacy classes, Fischer matrices and ordinary and projective tables of H 2 , we concluded that we only need to use the ordinary character table of H 2 to construct the character table of G 4. The Fischer matrices of G 4 are all listed in this paper. The character table of G 4 is a 195 × 195 complex valued matrix,...
1 a Characterization of Normal Subgroups via N-Closed Sets
2016
Let (G, *) be a semigroup, D ⊆ G, and n ≥ 2 be an integer. We say that (D, *) is an n-closed subset of G if a 1 * • • • * an ∈ D for every a 1 , ..., an ∈ D. Hence every closed set is a 2-closed set. The concept of n-closed sets arise in so many natural examples. For example, let D be the set of all odd integers, then (D, +) is a 3-closed subset of (Z, +) that is not a 2-closed subset of (Z, +). If K = {1, 4, 7, 10, ...} , then (K, +) is a 4-closed subset of (Z, +) that is not an n-closed subset of (Z, +) for n = 2, 3. In this paper, we show that if (H, *) is a subgroup of a group (G, *) such that [H : G] = n < ∞, then H is a normal subgroup of G if and only if every left coset of H is an n + 1-closed subset of G.
Verbal wreath products and certain product varieties of groups
Journal of the Australian Mathematical Society, 1967
Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.
On Groups with Two Isomorphism Classes of Derived Subgroups
Glasgow Mathematical Journal, 2013
The structure of groups which have at most two isomorphism classes of derived subgroups ($\mathfrak{D}$2-groups) is investigated. A complete description of mathfrakD\mathfrak{D}mathfrakD2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble mathfrakD\mathfrak{D}mathfrakD2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble mathfrakD\mathfrak{D}mathfrakD2-groups is found, and the locally free mathfrakD\mathfrak{D}mathfrakD2-groups are characterized.