International Journal of Group Theory
Volume:10 Issue: 4, Dec 2021

‎We show that for each positive integer $n$‎, ‎there exist a group $G$ and a subgroup $H$ such that the ordinary depth $d(H‎, ‎G)$ is $2n$‎. ‎This solves the open problem posed by Lars Kadison whether even ordinary depth larger than 6 can occur‎.

* The formulas are not displayed correctly.

There is a long-standing conjecture attributed to I. Schur that if G is a finite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. In this note I give an example of a four generator group G of order 5^{4122} with exponent 5, where the Schur multiplier M(G) has exponent 25.

* The formulas are not displayed correctly.

‎Let G be a finite group which is not cyclic of prime power order‎. ‎The join graph Δ(G) of G is a graph whose vertex set is the set of all proper subgroups of G‎, ‎which are not contained in the Frattini subgroup G and two distinct vertices H and K are adjacent if and only if G=⟨H‎,‎K⟩‎. ‎Among other results‎, ‎we show that if G is a finite cyclic group and H is a finite group such that Δ(G)≅Δ(H)‎, ‎then H is cyclic‎. ‎Also we prove that Δ(G)≅Δ(A5) if and only if G≅A5‎.

Let $n$ be a positive integer and let $G$ be a group‎. ‎We denote by $nu(G)$ a certain extension of the non-abelian tensor square $G otimes G$ by $G times G$‎. ‎Set $T_{otimes}(G) = {g otimes h mid g,h in G}$‎. ‎We prove that if the size of the conjugacy class $left |x^{nu(G)} right| leq n$ for every $x in T_{otimes}(G)$‎, ‎then the second derived subgroup $nu(G)''$ is finite with $n$-bounded order‎. ‎Moreover‎, ‎we obtain a sufficient condition for a group to be a BFC-group‎.

* The formulas are not displayed correctly.

‎Let $V$ be a unitary space‎. ‎Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$‎. ‎Consider the generalized symmetrizer on the tensor space $Uotimes V^{otimes m}$‎, ‎$$ S_{Lambda}(uotimes v^{otimes})=dfrac{1}{|G|}sum_{sigmain G}Lambda(sigma)uotimes v_{sigma^{-1}(1)}otimescdotsotimes v_{sigma^{-1}(m)} $$ defined by $G$ and $Lambda$‎. ‎The image of $Uotimes V^{otimes m}$ under the map $S_Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $Lambda$ and is denoted by $V_Lambda(G)$‎. ‎The elements in $V_Lambda(G)$ of the form $S_{Lambda}(uotimes v^{otimes})$ are called generalized decomposable tensors and are denoted by $ucircledast v^{circledast}$‎. ‎For any linear operator $T$ acting on $V$‎, ‎there is a unique induced operator $K_{Lambda}(T)$ acting on $V_{Lambda}(G)$ satisfying $$ K_{Lambda}(T)(uotimes v^{otimes})=ucircledast Tv_{1}circledast cdots circledast Tv_{m}‎. ‎$$ If $dim U=1$‎, ‎then $K_{Lambda}(T)$ reduces to $K_{lambda}(T)$‎, ‎induced operator on symmetry class of tensors $V_{lambda}(G)$‎. ‎In this paper‎, ‎the basic properties of the induced operator $K_{Lambda}(T)$ are studied‎. ‎Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions‎.

* The formulas are not displayed correctly.