Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [socle]

Questions relating to the direct product of minimal normal subgroups in a group or sum of all minimal nonzero submodules of a module.

Socle of a group is the subgroup generated by all minimal normal subgroups of a group, i.e. the join of all minimal normal subgroups

Socle of a module is the sum of all minimal nonzero submodules of a module.

Related with .

88 questions
6
votes
2 answers

How to understand the automorphism group of a very symmetric graph (related to sylow intersections)

For a group $G$ and subgroup $H$, consider the relation on $G$ defined $x \sim y$ if $H^x \cap H^y = 1$. This defines a graph on $G$. It is always fairly symmetric: $N_G(H)$ acts on the left and $G$ on the right as graph automorphisms. For some…
Jack Schmidt
  • 56,967
6
votes
1 answer

Socle of a direct product of finite groups.

The socle of a group $G$ is defined as the subgroup generated by minimal subgroups among normal subgroups of $G$, and it is denoted as $\textrm{Soc}(G)$. Suppose $A_1,...,A_n$ are finite groups. Is it true that $\textrm{Soc}(A_1 \times ... \times…
John P
  • 1,246
6
votes
0 answers

Definition of socle of a module

For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$. In the case of a local ring, I came across the…
Bubaya
  • 2,408
5
votes
1 answer

Socle of abelian divisible periodic group

I'm trying to prove that the socle of a periodic divisible abelian group J is a proper subgroup of J. I know that J is direct sum of quasicyclic groups, say $$ J={\oplus}_{i\in I} P_i $$ and that the socle of J is $$ Soc(J)={\oplus}_p J[p] $$ Now,…
Rob
  • 129
5
votes
1 answer

The socle of an almost simple group

By definition an almost simple group is a group $G$ with $$T \cong \mathrm{Inn}(T)\trianglelefteq G \leq \mathrm{Aut}(T),$$ where $T$ is a non-abelian simple group. How would one show that in this case the socle of $G$ is equal to…
Ishika
  • 397
5
votes
1 answer

Normal closure is minimal normal subgroup

The following problem is from the book "Finite Group Theory" by Martin Isaacs. (2.A.7) Let $S \lhd \lhd G$ (S is subnormal in G), where $S$ is nonabelian and simple and $G$ is finite. Show that $S^{G}$, the normal closure of $G$ is minimal normal…
Stefan4024
  • 36,357
4
votes
1 answer

Simple subgroups and the Wielandt subgroup

My goal is to prove the following: Let $G$ be a finite group and let $S \leq G$ be a simple subgroup. Suppose that $SH = HS$ for all subnormal subgroups $H$ of $G$. Show that $S$ is contained in the Wielandt subgroup $\omega(G)$ of $G$ Note:…
Gauss
  • 2,969
4
votes
1 answer

Doubt with Socle and O'Nan-Scott Theorem.

The following is the statement of O'Nan-Scott Theorem. Theorem: Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of G. Then either (a) $H$ is a regular elementary abelian $p$-group for some prime $p$, $n=p^m:=|H|$, and $G$…
Jins
  • 564
4
votes
1 answer

Finite non-abelian groups such that $Soc(G)=G$

Let $G$ be a finite group. It is possible that $Soc(G)=G$. For example: if $G=\prod_{i=1}^n \mathbb{Z}_{p_i}$ for some primes $p_1,\dotsc,p_n$ (not necessarily distinct), then $Soc(G)=G$. Can this happen for non-abelian groups? If the answer is…
user3533
  • 3,375
4
votes
1 answer

socles of semiperfect rings

For readers' benefit, a few definitions for a ring $R$. The left (right) socle of $R$ is the sum of all minimal left (right) ideals of $R$. It may happen that it is zero if no minimals exist. A ring is semiperfect if all finitely generated modules…
rschwieb
  • 160,592
4
votes
0 answers

(Reference Request) Socle of a module.

I came across the term "Socle" of a module defined for a finitely generated module $M$ over a noetherian ring $(A,\mathbb{m})$ as follows $$\mathrm{Soc}(M) = \lbrace x \in m \mid ax = 0\ \forall a \in \mathbb{m} \rbrace.$$ This definition is same as…
random123
  • 1,953
4
votes
1 answer

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then $(a+b)x=ax+bx$. As $ax\in A$ and $bx\in B$, this element is in…
man_in_green_shirt
  • 2,676
4
votes
1 answer

Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof any help would be great thanks
Peter A
  • 1,116
3
votes
1 answer

On one-dimensional socles

Let $(R,m,k)$ be a regular local ring of dimension $n$. Let $b_1,\dots,b_n$ be a maximal $R$-sequence and define $J=(b_1,\dots,b_n)$. Let $y_1,\dots,y_n$ be a regular system of parameters of $R$ and write $b_i = \sum b_{ji} y_j$. Then the…
Manos
  • 26,949
3
votes
2 answers

Socle of an algebra.

Let us consider the following sub algebra of $\mathbb{M}_4(K)$: \begin{equation} A=\begin{pmatrix} K & 0 & 0 & 0\\ K & K & 0 & 0\\ K & 0 & K & 0\\ K & K & K & K \end{pmatrix} \end{equation} Calculate soc($A)$ (as $A$-module). My attempt. By…
Ghost_Math125
  • 31
1
2 3 4 5 6