scholarly journals Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

scholarly journals Projective characters of degree one and the inflation-restriction sequence

1989 ◽  
Vol 46 (2) ◽  
pp. 272-280 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Schur Multiplier ◽  
Original Result ◽  
Image Position ◽  
A Finite Group ◽  
Invariant Element

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

Characteristic classes for permutation representations

1981 ◽  
Vol 90 (2) ◽  
pp. 265-272 ◽  
Author(s):  
G. B. Segal ◽  
C. T. Stretch
Keyword(s):  
Finite Group ◽  
Characteristic Classes ◽  
Cohomology Groups ◽  
Trivial Representation ◽  
Real Representation ◽  
Finite Dimensional ◽  
Whitney Class ◽  
Image Position ◽  
A Finite Group

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney classin the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and(b) stable, i.e. w(V) = 1 when F is a trivial representation.

scholarly journals On normal complements to sections of finite groups

1975 ◽  
Vol 19 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Everett C. Dade
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Subgroups ◽  
Normal Complement ◽  
Image Position ◽  
A Finite Group

Suppose that H/N is a section of a finite group G, i.e., that H is a subgroup of G and N is a normal subgroup of H. We are interested in the existence of normal subgroups M of G satisfying: Such an M can be called a normal complement to the section H/N in G.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

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