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.
645
questions
0
votes
1
answer
71
views
$[G:H]=4$ where $H\neq 1$ means that $G$ is not simple.
The question is as follows:
Suppose $G$ is a finite group with a nontrivial subgroup of index $4$. Prove that $G$ is not simple.
I am on the hunt for a non-trivial normal subgroup. This proof ...
4
votes
1
answer
109
views
Diagonal in the power of a group
Let $L$ be a simple group and $G=L^t$. The diagonal $D=\lbrace(x,x,\ldots,x), x\in L\rbrace$ is a subgroup of $L^t$. I can prove that if $D$ is maximal then $t$ is a prime.
If $t=mn$ then the subgroup ...
- 149
-1
votes
1
answer
50
views
Are commutator subgroups simple?
I am interested in the following setting:
$$ 0\rightarrow [G,G]\rightarrow G \rightarrow \mathbb{Z}^r\rightarrow 0, $$
in particular in the case $r=1$. Is $[G,G]$ simple in this case?
I am aware of ...
- 156
0
votes
0
answers
51
views
If $G$ is simple, then $Z({\rm Aut}(G)) = \{1\} \iff G$ is non-abelian [duplicate]
NOTE: I am aware of the link at For a Simple Group G, Z(Aut(G)) Is Trivial if and only if G is Non-Abelian. I am looking for proof verification of $G$ non-abelian $\implies Z(\mathrm{Aut}(G)) = \{1\}$....
- 1,788
1
vote
0
answers
233
views
"Simple" group of order $1004913$ problem, fixed point part
Let $G$ be a group of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$. We suppose that $G$ is simple. We want to obtain a contradiction.
This is the Exercise 29 in Chapter 6.2 of Dummit-Foote. As ...
- 405
1
vote
1
answer
46
views
A reference request for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$.
Note: This is a reference-request question and thus does not need the usual type of context.
The Question:
What is a reference for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$?
Background:
...
- 43.6k
6
votes
3
answers
441
views
Has every infinite simple group a faithful irreducible representation?
Has every infinite simple group a faithful irreducible representation?
This question solves the finite case. However, the proof requires a non-trivial linear representation of a finite group. I want ...
- 417
1
vote
0
answers
26
views
Splitting of an extension
Let $G$ be a group which is the extension of a free abelian group $A$ of finite rank by a finite simple group $S$. Does $G$ splits over $A$? (that is, $G=F\ltimes A$ for some finite subgroup $F\simeq ...
- 4,140
4
votes
0
answers
86
views
Motivation for triple cover of $A_6$ (and $A_7$)
In finite simple groups by Wilson, he constructs the triple cover of $A_6$ by considering the action of the subgroup of $A_6$ preserving the partition $\{12, 34, 56\}$ on the two vectors $(0, 0, 1, 1, ...
- 41
4
votes
0
answers
78
views
Is there a way to computationally verify that the sporadic groups are simple?
I'm trying to understand the "easy" direction of the CFSG: namely, the proofs that the 18 infinite families and 26/27 sporadic groups are indeed simple. I'm working through Simple Groups of ...
- 51
0
votes
1
answer
74
views
The only proper normal subgroups of a nonabelian quasisimple group are subgroups of its centre.
This is part of a bunch of exercises set by my academic supervisor. As such, I'm not sure whether all the hypotheses are needed for the conclusion. Please note that hints are preferred.
The Question:
...
- 43.6k
1
vote
1
answer
60
views
Does every perfect finite group have a fixed point free representation?
Fixed point free representations of finite group are important for the spherical space form problem and also show up in other contexts for example Perfect semi direct products
A representation $ \pi $ ...
- 7,124
2
votes
2
answers
176
views
Show $G$ is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$
Let $G$ be a finite group with at least two (distinct) subgroups of index $2$, and suppose that at least one of the index-$2$ subgroups of $G$ is simple. Prove that $G\cong \mathbb{Z}_2 \times \mathbb{...
- 115
2
votes
1
answer
84
views
Is there an explicit maximal simple group?
It is not hard to prove the following lemma:
LEMMA: Let $(G_i)_{i \in I}$ be a chain of simple groups.
Then $G = \bigcup_{i \in I} G_i$ is a simple group.
Let $N$ be a normal subgroup of $G$ and ...
- 316
0
votes
1
answer
58
views
Show that $ SL(2,5) $ has no subgroup isomorphic to $ A_5 $
I'm trying to show that there's no subgroup of $ SL(2,5) $ isomorphic to $ A_5 $. I've already shown that $ A_5 $ is simple, and my strategy is to show that $ SL(2,5) $ contains no simple subgroups of ...
- 65
3
votes
1
answer
71
views
Finite non abelian simple groups whose order is not divisible by 8
Is $A_5$ the only finite non abelian simple group whose order is not divisible by 8?
If not, what is the complete set of finite non abelian simple groups whose order is not divisible by 8?
- 352
0
votes
0
answers
38
views
Simplicity of the $A_5$ [duplicate]
In the proof given in Abstract Algebra by Dummit and Foote, page 128, it states that $(1~2)(3~4)$ commutes with $(1~3)(2~4)$ but does not commute with any element of odd order in $A_5$. So it follows ...
- 9
4
votes
1
answer
138
views
Let $G$ be a finite simple non abelian group and $\{1\}\neq H\le G$ be such that $\vert C_{G}(H)\vert=\vert G:H \vert$. Then $H=G$.
Let $G$ be a finite simple non abelian group and $\{1\}\neq H\leq G$ be such that $\vert C_{G}(H)\vert=\vert G:H \vert$. Show that $H=G$.
I can find only $[G,G]=G$ and $Z(G)=\{1\}$ and if $H$ is a ...
- 2,423
8
votes
1
answer
191
views
Is every finite simple group a quotient of a braid group?
Question: Is every finite simple group a quotient of a braid group?
Context: The braid group on two strands $ B_2 $ is isomorphic to $ \mathbb{Z} $ and so the infinite family of abelian finite simple ...
- 7,124
2
votes
1
answer
127
views
There are no simple groups of order $480$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is 480; this self-answered question aims to fill that gap. (...
- 94.7k
2
votes
1
answer
256
views
There are no simple groups of order $336$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $336$; this self-answered question aims to fill that gap....
- 94.7k
3
votes
1
answer
141
views
There are no simple groups of order $264$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $264$; this self-answered question aims to fill that gap....
- 94.7k
0
votes
0
answers
40
views
a doubt on the simplicity of the alternating group $A_n$ in Dummit&Foote's Abstract Algebra
I have a doubt about the simplicity of the alternating group $A_n$ in Dummit&Foote's Abstract Algebra on page150 :
It's to prove that the alternating group $A_n=G$ is simple for $n\geq5$ . Let $H\...
- 115
4
votes
1
answer
103
views
To what extent can we determine the simplicity or non-simplicity of groups based on their prime decompositions?
This question may have more of a vague, less objective answer than usual for this site, so I apologise if it difficult to answer definitively.
Below, $p$, $q$ and $r$ are distinct primes.
A group of ...
- 1,739
4
votes
0
answers
105
views
Every proper subgroup of a simple group containing an order 45 element has an index of at least 14
Every proper subgroup of a simple group containing an order $45$ element has an index of at least $14$.
So far I've supposed that $G$ is a simple group with an element of order $45$. Then $45$ ...
- 93
0
votes
1
answer
86
views
Automorphisms of $ SL(n,q) $ coming from $ GL(n,q) $
For which values of $ n \geq 2 $ and $ q $ a prime power are all automorphisms of $ SL(n,q) $ induced by conjugation by elements of $ GL(n,q) $?
Old version of the question:
$ \Sigma L(n,q) $ is an ...
- 7,124
0
votes
1
answer
88
views
Natural group extension constructed from Schur cover and its outer automorphism group
Let $ S $ be a finite (non-abelian) simple group. Then there always exists a natural extension of $ S $ by the outer automorphism group $ Out(S) $ with elements of $ Out(S) $ acting as outer ...
- 7,124
1
vote
0
answers
71
views
Nontrivial extension of cyclic group by simple group
Let $ G $ be a (non-abelian) finite simple group. An extension $ G\cdot m $ is nontrivial if it is not isomorphic to the direct product $ G \times m $. Suppose that there exists a nontrivial extension ...
- 7,124
5
votes
3
answers
302
views
No simple group of order 1040
Burnside (Proceedings of the London Mathematical Society, vol. 26; Collected Papers, vol. 1, p. 601) gave the following proof of the non-simplicity of groups od order 1040 :
"If simple, the group ...
- 1,675
1
vote
1
answer
219
views
Show that a finite simple group $G$ of even order must have order divisible by $4$.
It is requested to proof exercise $3.30$, $(ii)$ from J. Rotman, An introduction to the theory of groups, which states that a finite simple group $G$ of even order greater than $4$ must have order ...
- 719
3
votes
0
answers
63
views
Equivalent of simple groups in topology
I read about the classification theorem of finite simple groups, and I was wondering if there is a topological meaning to spaces that have a simple fundamental group?
In addition, is there something ...
- 115
0
votes
1
answer
106
views
On Subgroups of Same order of Simple group
We know that, If there exists none other than only subgroup $H$ of order $m$, then $H$ is normal in $G$. When It comes to simple group, There should exist at the least $2$ subgroups of same order. To ...
user834772
1
vote
1
answer
22
views
Image of minimal degree representation of quasisimple group unique up to conjugacy.
Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of ...
- 7,124
1
vote
2
answers
127
views
What is the relation of word "simple" with math-word "simple group" of group theory?
Is there any relation with the meaning of word "simple" with what "there are groups $G$ in which the only normal subgroups are the trivial ones: $1$ and $G$. Such groups are called ...
- 915
5
votes
1
answer
222
views
Is every finite non-abelian simple group generated by involutions?
Let $ G $ be a finite non-abelian simple group. Is it true that the set of involutions
$$
\{ g: g\in G, g^2=1 \}
$$
generates $ G $?
For example consider the group $ G=A_5 $ of order $ 60 $. The ...
- 7,124
9
votes
0
answers
155
views
Why is studying centralizers the/a key to classifying finite groups?
In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says
One problem, as least with the current methods of ...
- 8,363
11
votes
0
answers
325
views
Why do we trust the Classification of Finite Simple Groups?
It seems to me there are a two main reasons to believe a theorem/conjecture to be true:
Because it has a correct proof (e.g. the Feit-Thompson Theorem, Dirichlet's Theorem)
Because there is an ...
- 4,074
0
votes
1
answer
200
views
Let $G$ be a simple group. Show that any homomorphism from $ G$ to $G'$ (arbitrary $G'$) must be either injective or the trivial homomorphism. [duplicate]
I've spent a lot of time thinking about this question but I can't seem to come up with anything of substance. I tried to use some basic lemma for group homomorphisms, but nothing seemed to work. My ...
- 245
1
vote
1
answer
49
views
Simplicity of a subgroup of $S_p$
The following is an exercise, to prove simplicity of a group $G$, which satisfies conditions -
i) $G$ is transitive subgroup of $S_p$ (symmetric group, $p$ is prime)
ii) $|G|=p\cdot m\cdot k$,
iii) $m\...
- 2,777
0
votes
0
answers
50
views
Statement explanation of a paper about finite simple groups
I was trying to understand the statement of the main theorem of this paper https://arxiv.org/pdf/1510.03665.pdf The statement says that if $S$ is a finite non-abelian simple group and $r$ is an odd ...
- 4,140
0
votes
1
answer
48
views
Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
- 7,124
1
vote
1
answer
72
views
Group mod center is perfect structure result
Suppose that
$$
1 \to Z(G) \to G \to P \to 1
$$
is a short exact sequence of groups where $ P $ is perfect and $ Z(G) $ is the center of $ G $. Must it be the case that $ G $ is the direct product of ...
- 7,124
2
votes
1
answer
151
views
A question about the classification of compact Lie groups
I was able to prove that every compact Lie group is isomorphic to
$$(T^n\times G_1\times \cdots \times G_m)/\Gamma$$
where the $G_i$ are compact, simply connected, simple Lie groups and $\Gamma$ is a ...
- 1,047
2
votes
1
answer
169
views
Prove that $A_5$ is simple.
Prove that $A_5$ is simple.
The solution presented is as follows:
Let $N$ be a non trivial normal subgroup of $A_5.$ If $(12)(34)\in N,$ then $$(25)(34)(12)(34)(34)(25) = (15)(34)$$ and $$(12)(34)(15)...
- 2,564
0
votes
1
answer
113
views
For $n\ge 5,\forall e\neq\rho\in A_n,\exists\sigma=\alpha\rho\alpha^{-1}$ for some $α\in A_n$ s.t. $ρ(i)=σ(i)$ for some $i\in\{1,2,...,n\}$, but $ρ≠σ$
For $n\ge 5$, for all $e\neq \rho\in A_n, \exists \sigma=\alpha\rho\alpha^{-1}$ for some $\alpha\in A_n$ such that $\rho(i)=\sigma(i)$ for some $i\in\{1,2,...,n\}$, but $\rho\neq\sigma$.
The solution ...
- 2,564
2
votes
0
answers
59
views
Is every finite group a subgroup of this type of generalization of almost simple groups?
Recall, a group $G$ is an almost simple group if there is an non-abelian simple group $S$ such that $S \leq G \leq \operatorname{Aut}(S)$
We will define a $k$-almost simple as follows.
When $0$--...
- 1,683
2
votes
0
answers
55
views
random generation of lie algebras
It is well known that a nonabelian finite simple group, say $\mathrm{PSL}_n(\mathbf{F}_p)$, can be generated by two elements. In fact, the probability that two elements generate it tends to $1$ as the ...
- 1,191
1
vote
1
answer
107
views
Why can't a group of order 132 contain 3 Sylow $2$-subgroups and be simple?
In the question titled Prove that if |G|=132 then G cannot be simple, it is shown a group of order $132$ cannot be simple. My summary of the proof:
assume group is simple
deduce the number of Sylow $...
- 995
2
votes
1
answer
53
views
Minimal normal subgroups of normal closure
This question arose from my attempt at understanding the answer in this post - the comments below it, to be precise. Everything revolves around the following problem:
Let $S$ be a simple, nonabelian ...
- 2,289
4
votes
1
answer
104
views
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 ...
- 2,289