The Noether number of p-groups

A group of order [Formula: see text] ([Formula: see text] prime) has an indecomposable polynomial invariant of degree at least [Formula: see text] if and only if the group has a cyclic subgroup of index at most [Formula: see text] or it is isomorphic to the elementary abelian group of order 8 or the Heisenberg group of order 27.

CI-Property for Decomposable Schur Rings over an Abelian Group

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

FITTING SUBGROUP AND NILPOTENT RESIDUAL OF FIXED POINTS

Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

Realizability and the Avrunin–Scott theorem for higher-order support varieties

We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin–Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let q be a prime, n a positive integer and A an elementary abelian group of order {q^{r}} with {r\geq 2} acting on a finite {q^{\prime}} -group G. We show that if all elements in {\gamma_{r-1}(C_{G}(a))} are n-Engel in G for any {a\in A^{\#}} , then {\gamma_{r-1}(G)} is k-Engel for some {\{n,q,r\}} -bounded number k, and if, for some integer d such that {2^{d}\leq r-1} , all elements in the dth derived group of {C_{G}(a)} are n-Engel in G for any {a\in A^{\#}} , then the dth derived group {G^{(d)}} is k-Engel for some {\{n,q,r\}} -bounded number k. Assuming {r\geq 3} , we prove that if all elements in {\gamma_{r-2}(C_{G}(a))} are n-Engel in {C_{G}(a)} for any {a\in A^{\#}} , then {\gamma_{r-2}(G)} is k-Engel for some {\{n,q,r\}} -bounded number k, and if, for some integer d such that {2^{d}\leq r-2} , all elements in the dth derived group of {C_{G}(a)} are n-Engel in {C_{G}(a)} for any {a\in A^{\#},} then the dth derived group {G^{(d)}} is k-Engel for some {\{n,q,r\}} -bounded number k. Analogous (non-quantitative) results for profinite groups are also obtained.

The non-solvable doubly transitive dimensional dual hyperovals

In [9] S. Yoshiara determines possible automorphism group of doubly transitive dimensional dual hyperovals. He shows that a doubly transitive dual hyperovalDis either isomorphic to the Mathieu dual hyperoval or the dual hyperoval is defined over 𝔽2, and if the hyperoval has rankn, the automorphism group has the formE⋅S, with an elementary abelian groupEof order 2nandSa subgroup of GL(n,2) acting transitively on the nontrivial elements ofE. Moreover Yoshiara describes the possible candidates forS. In this paper we assume thatSis non-solvable and show that then the dimensional dual hyperoval is a bilinear quotient of a Hyubrechts dual hyperoval.

Locally 3-Arc-Transitive Regular Covers of Complete Bipartite Graphs

In this paper, locally $3$-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally $3$-arc-transitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally $3$-arc-transitive regular covers of complete bipartite graphs with covering transformation group isomorphic to a cyclic group or an elementary abelian group of order $p^2$.

DERIVED SUBGROUPS OF FIXED POINTS IN PROFINITE GROUPS

The main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that CG(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

Transform Domain Characterization of Dual Group Codes of Cyclic Group Codes over Elementary Abelian Groups

The group of characters of an elementary Abelian group has been used to define duality between its subgroups, which in turn is extended to duality between group codes. The transform domain description of the dual codes of cyclic group codes of length over has been developed in this paper. Several example codes and their duals have been presented also.