Questions tagged [simple-groups]
Use with the (group-theory) tag. A group is simple if it has no proper, non-trivial normal subgroups. Equivalently (for finitely generated groups), its only homomorphic images are itself and the trivial group. The classification of finite simple groups is one of the great results of modern mathematics.
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No simple group of order 756 : Burnside's proof
I'm interested in a proof of the non-simplicity of groups of order 756. W.R. Scott, Group Theory, p. 392, exerc. 13.4.9, gives it as an easy exercise, but depending on rather advanced results.
I have ...
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"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?
I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
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Use of correspondence theorem for groups to prove that $o(N) = 2$
Let $G$ be a group and $H \triangleleft G$ simple such that $[G : H] = 2$. I have to prove that if $N \neq \{1\}$, $N \triangleleft G$ and $N \cap H = \{1\}$ then $o(N) = 2$.
I know by third ...
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Proof that a group of order $180$ is not simple without Burnside p-complement theorem
A proof that a group of order $180$ is not simple is given here.
However, the proof uses Burnside $p$-complement theorem.
If you know a proof without Burnside $p$-complement theorem, please let me ...
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0
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On the number of invariant Sylow subgroups under coprime action - Antonio Beltrán and Changguo Shao article
This is an article that Antonio Beltrán and Changguo Shao wrote. Lemma 2.5. states: [All groups are supposed to be finite (this is mentioned before)]
Lemma 2.5. Let $A$ be a group acting coprimely on ...
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1
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A question on an isomorphism between ${\rm PSL}_2(9)$ and $A_6$ [closed]
I found a nice argument proving that ${\rm PSL}_2(9)\cong A_6$ on page 52 of The finite simple groups by Prof. R.A. Wilson.
Let $f:A_6\to S_{{\rm PL}(9)}$ with $(123)^f= z\mapsto z+1$, $(456)^f= z\...
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Probability that two elements commute in a noncommutative simple finite group
Good afternoon !
Let $G$ a finite non-abelian group. Let $p_G$ the probability that two elements randomly chosen commute. It is well known that : $$p_G \leqslant \frac{5}{8}$$
The upper bound is ...
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1
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Automorphism group of ${\rm PSL}_2(p)$ [closed]
This question is related to this answer by Prof. Holt.
I can see why ${\rm PGL}_2(p)$ induces the full automorphism group of a Sylow $p$-subgroup $S$ of ${\rm PSL}_2(p)$. Let $a$ be any automorphism ...
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Which alternating groups are characteristic 2 type?
First, let's recall the definition of characteristic 2 type (I think this is correct, but I am not 100% sure). For any 2-local subgroup $H$ of $G$, if $H$ contains the entire Sylow 2-subgroup, all ...
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Problem 5C.3 Isaacs' Finite Group Theory
I have a question about the following problem [Finite Group Theory, Martin Isaacs, Chapter 5]:
Let $G$ be simple and have an abelian Sylow 2-subgroup $P$ of order $2^{5}$. Deduce that $P$ is ...
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Groups of order $2^n p$ for $n\geq 1$ and $p$ prime with $2^n> (p-1)!$ are non-simple. Is my proof correct?
I'm doing my homework in Group Theory and as part of an exercise, I want to show the following Lemma:
Let $n\geq 1$, $p$ a prime, s.t. $2^n > (p-1)!$ and $G$ a group of order $2^n p$. Then G has a ...
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Show $\mathrm{Inn}(G)\,\operatorname{char}\,\mathrm{Aut}(G)$ for $G$ a non-abelian simple group
Let $G$ a non-abelian simple group and let $A=\mathrm{Inn}(G)$ and $B=\mathrm{Aut}(G)$.
I would like to know the solution to $A\,\operatorname{char}\, B$.
However, I know the following. Let $\phi \in \...
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votes
1
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Prove the index of a proper subgroup of a simple group of order 17971200 is at least 14.
I didn't find a solution for this problem or other usual approaches that could directly work. So, here is my attempt. I am self-studying and reviewing group theory recently, and would like to know if ...
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Isomorphic two abelian subgroups which lies in the some finite union of conjugacy classes of simple group [closed]
Let $G$ be a simple group and $H_1$ and $H_2$ be abelian subgroups of $G$ such that $H_1\cong H_2$ and $H_1,H_2\subseteq Cl_G(e_G)\cup Cl_G(x)$ for some $x\in G$, where $Cl_G(\cdot)$ denotes the ...
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"$S$ is the unique irreducible $\operatorname{End}(S)$-module" [duplicate]
Let $S$ be a finite-dimensional vector space. Then $S$ is an irreducible $\operatorname{End}(S)$-module. Furthermore I was told that $S$ is the only irreducible $\operatorname{End}(S)$-module. I guess ...
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