scholarly journals The unitary cover of a finite group and the exponent of the Schur multiplier

2015 ◽  
Vol 426 ◽  
pp. 344-364 ◽  
Author(s):  
Nicola Sambonet
Keyword(s):  
Finite Group ◽  
Schur Multiplier ◽  
A Finite Group

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.

scholarly journals A NOTE ON p-CENTRAL GROUPS

2013 ◽  
Vol 55 (2) ◽  
pp. 449-456
Author(s):  
RACHEL CAMINA ◽  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Prime Number ◽  
Nilpotency Class ◽  
Schur Multiplier ◽  
Tate Cohomology ◽  
Explicit Bound ◽  
Covering Group ◽  
A Finite Group

AbstractA group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

Real projective representations of finite groups

1990 ◽  
Vol 107 (1) ◽  
pp. 27-32 ◽  
Author(s):  
P. N. Hoffmann ◽  
J. F. Humphreys
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Schur Multiplier ◽  
Representations Of Finite Groups ◽  
Projective Representations ◽  
A Finite Group

The projective representations of a finite group G over a field K are divided into sets, each parametrized by an element of the group H2(G, Kx). The latter is the Schur multiplier M(G) when K = ℂ.

scholarly journals On the multiple exterior degree of finite groups

2014 ◽  
Vol 64 (4) ◽  
Author(s):  
Rashid Rezaei ◽  
Peyman Niroomand ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Schur Multiplier ◽  
Group Invariant ◽  
Exterior Degree ◽  
A Finite Group ◽  
Exterior Square

AbstractRecently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements x and y of a finite group G such that xΛy = 1 in the exterior square GΛG of G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups.

scholarly journals Subgroups of the Schur multiplier

1990 ◽  
Vol 48 (3) ◽  
pp. 497-505 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Schur Multiplier ◽  
Odd Order ◽  
A Finite Group

AbstractTwo subgroups ME(G) and MI(G) of the Schur multiplier M(G) of a finite group G are introduced: ME(G) contains those cohomology classes [α] of M(G) for which every element of G is α-regular, and MI(G) consists of those cohomology classes of M(G) which contain a G-invariant cocycle. It is then shown that under suitable circumstances, such as when G has odd order, that each element of MI(G) can be expressed as the product of an element of ME(G) and an element of the image of the inflation homomorphism from M(G/G′) into M(G).

NOETHER SETTINGS FOR CENTRAL EXTENSIONS OF GROUPS WITH ZERO SCHUR MULTIPLIER

2002 ◽  
Vol 01 (01) ◽  
pp. 107-112 ◽  
Author(s):  
ESTHER BENEISH
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cyclic Group ◽  
Central Extension ◽  
Closed Field ◽  
Schur Multiplier ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
Rational Extension ◽  
A Finite Group

Let G be a finite group, and let F be an algebraically closed field. The rational extension of F, F(G) = F(xg : g ∈ G) with G action given by hxg = xhg for g, h ∈ G, is referred to as the Noether setting for G. We show that if G is a group with zero Schur multiplier, then for any central extension G′ of G, F(G′)G′ and F(G)G are stably equivalent over F. That is, F(G)G is stably rational over F if and only if F(G′)G′ is stably rational over F. In particular, if G is a cyclic group or an abelian group with zero Schur multiplier, then F(G′) is stably rational over F.

scholarly journals The efficiency of PSL(2, p)3 and other direct products of groups

1997 ◽  
Vol 39 (3) ◽  
pp. 259-268 ◽  
Author(s):  
C. M. Campbell ◽  
I. Miyamoto ◽  
E. F. Robertson ◽  
P. D. Williams
Keyword(s):  
Finite Group ◽  
Direct Products ◽  
Schur Multiplier ◽  
Converse Problem ◽  
Number Of Generators ◽  
Products Of Groups ◽  
A Finite Group ◽  
Finite Presentations ◽  
Minimum Deficiency

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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