# 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.

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### Artinian module's socle is essential

How do I prove that the socle of an Artinian module is an essential submodule? I don't see where we should use the artinianity of the module to prove this.
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### Finding a form of Representations of algebra $FV_4$, where $F=Z_2$

I am hoping for some help with the following exam question I attempted. It is as follows, broken in to two parts: Let $A$ be a finite-dimensional algebra over a field $F$. $1.)a)i)$ What is a ...
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### A subring of the upper triangular matrices

I want to compute the Jacobson radical, the right socle, and the left socle of the ring of $3\times 3$ upper triangular matrices with entries in $\mathbb Z_4$ and such that the entries on the main ...
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### Whether a non-zero module can have a zero socle?

Let $M$ be a module. Then $soc(M)=\sum\{N\leq M| \text{$N$is a simple submodule of$M$}\}=\cap\{L\leq M| \text{$L$is essential in$M$}\}$. I don't know whether a non-zero module can have zero socle?...
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### Decomposition of right ideals

Let $R$ be a unital ring with right socle $Soc(R_R)$ such that $R/Soc(R_R)$ is right weakly regular, i.e all whose right ideals are idempotent. Is it true that every right ideal $I$ of $R$ decomposes ...
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### Question concerning cyclic modules and modules where the socle of $M$ is irreducible

Let $R$ be a ring and $M$ be an $R$-module of finite length $n$ (given by the number of composition factors). I am trying to prove or disprove: If $\mathrm{Soc}(M)$ (the socle of $M$) is a unique ...
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### Socle of a primitive permutation group

So we need to prove that the socle of a primitive permutation group is a direct product of isomorphic simple groups. Now socle means product of the minimal normal subgroups. I know that every non-...
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### 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 ...
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### Showing the socle of a module exists and is unique

I have been set the following exercise: For every $R$-module $M$, show that there exists a unique semi-simple submodule $sM$ $\subset$ $M$ which contains every semi-simple submodule of $M$. After ...
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### 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 ...
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### If a group 's socle is $Sz(q)$, how can I determine its maximal subgroup

As the title says,I want to know the maximal subgroup of the group whose socle is $Sz(q)$. If there are such papers,could you tell its name or give me a link. Thank a lot
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### 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 $G$ ...
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### 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 ...
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Let $M$ be a module of a finite dimensional algebra $A$ over an algebraically closed field $K$. Let $N=M/\operatorname{rad}M$ be the top of $M$ and suppose that $N$ is simple. Let $D=\hom_K(\cdot, K)$....
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### 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 ...
### Maximal subgroups of almost simple groups with socle $PSL(2, q)$
Let $G$ be an almost simple group with socle $PSL(2,q)$ where $q=p^f>3$ is the $f$th power of some odd prime $p$, and $M$ a maximal subgroup of $G$. By http://arxiv.org/abs/math/0703685, except for ...