A complete set of invariants for finite groups and other results

Moshe Roitman

doi:10.1016/0001-8708(81)90023-2

A complete set of invariants for finite groups and other results

Moshe Roitman

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For a finite group G and a set I ⊆ {1, 2,..., n} let _G(n,I) = ∑_{g ∈ G} ε₁(g)⊗ε₂(g)⊗⋯⊗ε_n(g),. where ε_i(g)=g if i=∈ I,ε_l(g)=l if i=∈ I. We prove, among other results, that the positive integers tr (e_G(n,I₁)+⋯+e_G(n,I_r))^k:n,r,k,≥1, I_j⊆{1,...,n}, 1≤|i_j|≤3. for 1 ≤ j ≤ r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ≤ j₁ < j₂ < j₃ < j₄ ≤ r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.

Original language	English
Pages (from-to)	301-311
Number of pages	11
Journal	Advances in Mathematics
Volume	41
Issue number	3
DOIs	https://doi.org/10.1016/0001-8708(81)90023-2
State	Published - Sep 1981

ASJC Scopus subject areas

General Mathematics

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10.1016/0001-8708(81)90023-2

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title = "A complete set of invariants for finite groups and other results",

abstract = "For a finite group G and a set I ⊆ {1, 2,..., n} let G(n,I) = ∑g ∈ G ε1(g)⊗ε2(g)⊗⋯⊗εn(g),. where εi(g)=g if i=∈ I,εl(g)=l if i=∈ I. We prove, among other results, that the positive integers tr (eG(n,I1)+⋯+eG(n,Ir))k:n,r,k,≥1, Ij⊆{1,...,n}, 1≤|ij|≤3. for 1 ≤ j ≤ r, Ij1 ∩ Ij2 ∩ Ij3 ∩ Ij4 = {\O} for any 1 ≤ j1 < j2 < j3 < j4 ≤ r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.",

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AU - Roitman, Moshe

PY - 1981/9

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N2 - For a finite group G and a set I ⊆ {1, 2,..., n} let G(n,I) = ∑g ∈ G ε1(g)⊗ε2(g)⊗⋯⊗εn(g),. where εi(g)=g if i=∈ I,εl(g)=l if i=∈ I. We prove, among other results, that the positive integers tr (eG(n,I1)+⋯+eG(n,Ir))k:n,r,k,≥1, Ij⊆{1,...,n}, 1≤|ij|≤3. for 1 ≤ j ≤ r, Ij1 ∩ Ij2 ∩ Ij3 ∩ Ij4 = Ø for any 1 ≤ j1 < j2 < j3 < j4 ≤ r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.

AB - For a finite group G and a set I ⊆ {1, 2,..., n} let G(n,I) = ∑g ∈ G ε1(g)⊗ε2(g)⊗⋯⊗εn(g),. where εi(g)=g if i=∈ I,εl(g)=l if i=∈ I. We prove, among other results, that the positive integers tr (eG(n,I1)+⋯+eG(n,Ir))k:n,r,k,≥1, Ij⊆{1,...,n}, 1≤|ij|≤3. for 1 ≤ j ≤ r, Ij1 ∩ Ij2 ∩ Ij3 ∩ Ij4 = Ø for any 1 ≤ j1 < j2 < j3 < j4 ≤ r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.

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A complete set of invariants for finite groups and other results

Abstract

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