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 ...

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3
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1answer
57 views

Simple groups with cyclic odd Sylow subgroups

I know that every odd Sylow subgroups of $PSL(2,p)$ is cyclic. Is there any other simple group with cyclic odd Sylow subgroups. Thank you in advance
0
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0answers
22 views

Some things about maximal tori

Let $G$ be a linear algebraic group over an algebraic closed field of characteristic $p\neq 2$. Suppose $\overline{G}_{\sigma}={G}_{\sigma}/Z({G}_{\sigma})$ where ${G}_{\sigma}$ is the set of fixed ...
2
votes
3answers
96 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
0
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0answers
60 views

Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
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0answers
21 views

Using a theorem to find the center of a $p$-sylow subgroup of simple group

I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown ...
0
votes
1answer
30 views

The center of Sylow $p$-subgroups of a finite simple group of Lie type

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of ...
2
votes
2answers
66 views

Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
1
vote
1answer
27 views

regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
1
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1answer
45 views

Simple non-abelian groups

Let $G$ be a group and $H$ be a simple non-abelian subgroup of $G$ which is ascendant in $G$. Is it true that $H$ is also subnormal in $G$? Definition Let $G$ be a group and $H$ be a subgroup of $G$. ...
2
votes
1answer
40 views

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$ Hint: consider the action of $G$ on right cosets of $H$ in $G$. I'm ...
4
votes
2answers
72 views

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
2
votes
1answer
45 views

A question about the involution in simple groups.

Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question ...
3
votes
1answer
48 views

How can I prove that $A_5$ is perfect?

I'm trying to prove that $A_5$ is perfect. The only proof I found until now is: "It follows from the fact that it's simple and non-abelian". Simplicity is quite stronger, and since I only need ...
1
vote
1answer
61 views

Check a proof for a theorem about the number of normal subgroups of $G \times H$ when $G$ and $H$ are simple

I'm going to provide a detailed proof for this theorem: Let $G$ and $H$ be non-trivial simple groups. If there is a prime number $p$ with $$|G|=|H|=p$$ then $G\times H$ has $p+3$ normal ...
0
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1answer
61 views

Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
0
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0answers
24 views

Existance of regular semisimple elements in a torus

I had asked a question about regular semisimple elements in finite simple group of Lie type in this link Existance of semisimple elements in a torus. Actually I wanted to know that is it true that ...
-1
votes
1answer
43 views

Why projective linear spaces are simple ??? [closed]

In classical groups, people project all those linear groups and make it as simple. Why do they so, Why can't linear groups can be restricted to simple instead of projecting on the scalar space. What ...
4
votes
1answer
76 views

Sort-of-simple non-Hopfian groups

A finite simple group is one which has no homomorphic images apart from itself and the trivial group. However, the simple-groups tag does not include the condition "finite". My question is the ...
1
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1answer
61 views

Size of maximal tori in finite simple groups of Lie type

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$? ...
0
votes
1answer
84 views

QUESTION about a regular dodecahedron [closed]

Can someone help me with this question? I have no clue. Thank you so much! Let G denote the group of rotational symmetries of a regular dodecahedron. This problem invloves considering the action of G ...
3
votes
3answers
104 views

Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
5
votes
1answer
70 views

Criterion for being a simple group

In this work it's written that A group $G$ is simple if and only if the diagonal subgroup of $G \times G$ is a maximal subgroup. How can I prove it?
1
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1answer
52 views

Number of Sylow $5$-subgroups

Suppose $G=SL(2,4)$. Could you please suggest, is there is any simple argument to show that $G$ cannot have the unique Sylow $5$-subgroup, i.e. $|Syl_5(G)|>1$ without quoting the isomorphism $A_5 ...
1
vote
1answer
34 views

Simple subgroup of transitive group

Suppose $Q \in Syl_n(G)$ and $Q$ is not normal in $G$, $Q$ is generated by an element of order $n$, where $G \leqslant S_n$ for prime $n$ and $G$ acts transitively on $\{1,...,n\}$. Define H to be a ...
2
votes
1answer
59 views

How to show that group of order 760 is not simple? [closed]

How to show that group of order 760 is not simple? By Sylow's theorem $n_{19}=20$, and $o(N(P)) = 38$, but how to continue after this? Thanks for any help
2
votes
1answer
47 views

Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
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votes
0answers
38 views

Existance of semisimple elements in a torus.

Let $G$ be a finite simple group of Lie type and $T$ be a maximal torus in $G$. Is it true that $T$ contains a regular semisimple element (a semisimple element which $C_G(x)=T$)? If yes, why?
5
votes
2answers
75 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
0
votes
0answers
19 views

dimension of a finie simple group of Lie type.

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$, namely $F_q$ of size $q$. Suppose $dim(G)=n$. What we can say about $|G|$? Is it true that $|G|=q^ns$, where ...
0
votes
1answer
75 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
0
votes
0answers
32 views

Brauer characters of finite simpl groups of type E8

I would like to know the Brauer characters of finite simple groups $E_8(2)$, $E_8(3)$ or $E_8(5)$. Is there any refrence for this topic? Thanks
4
votes
1answer
35 views

Automorphisms of spin groups over finite fields, even dimension

I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of ...
3
votes
1answer
87 views

Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
3
votes
1answer
73 views

Is $\mathrm{PSL} ( 2, \mathbb{Q} )$ a simple group?

I am a new poster but I don't think this question has been asked before. Pardon me if it is.
3
votes
2answers
133 views

Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy ...
0
votes
1answer
44 views

A question about nonabelian finite simple groups

Let $G$ be a nonabelian finite simple group of lie type on finite field $F$ and $s\in G$ be a semisimpl element of $G$, $i.e.$ an element with order coprime to $Char(F)$. Also suppose $T$ is a maximal ...
6
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1answer
103 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
2
votes
2answers
40 views

Are there any nontrivial ways to factor n-cycles into a product of cycles?

I was reading a proof here about the simplicity of $A_n (n \ge 5)$. It states (and proves) a lemma about 3-cycles: A 3-cycle $(a, b, c)$ may be written as $(a, b, c) = (1, 2, a)^{-1}(1, 2, c)(1, 2, ...
3
votes
0answers
64 views

Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
3
votes
2answers
136 views

Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
3
votes
2answers
156 views

An exercise about simple groups

Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...
1
vote
2answers
149 views

Prove that the only homomorphism between a simple non-abelian group G and abelian group A is trivial

Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial. OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})
4
votes
1answer
126 views

Prove that $G=S_{1}\times S_{2}\times\cdots\times S_{k} $

Let $G$ be a finite group and let $N_1,N_2,\ldots,N_k$ be normal subgroups. Suppose that $\bigcap_{i=1}^{k}N_{i}=\{1\}$ and that $G/N_i=S_i$ is simple. If all the groups $S_i$ are non-isomorphic ...
3
votes
1answer
76 views

Conjugacy classes of two elements of order $p$ in a simple group

Let $G$ be a non abelian simple group and $x,~y$ be two elements of order $p$ of $G$, where $p$ is a prime. Suppose $|x^G|\neq |y^G|$. Is there any relation between $|x^G|$ and $|y^G|$? For example is ...
3
votes
0answers
53 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
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votes
2answers
93 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
2
votes
2answers
86 views

Groups of order $56$

Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove? My idea: ...
3
votes
2answers
76 views

Proving that infinite union of simple groups is also simple group

Given that: $$G_1\subseteq G_2\subseteq G_3\subseteq \dots\subseteq G_n \subseteq G_{n+1}\subseteq \cdots$$ Are all simple groups. Prove that $$G=\bigcup_{n=1}^{\infty}G_n$$ is also a simple group. ...
2
votes
1answer
46 views

No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
0
votes
1answer
82 views

No group of order 2907 is simple

Show that any Group of order 2907 is not a simple group? 2907= 3*3*17*19 I've started with the Sylow 19-subgroup, then the 17-subgroups and finally the 3-subgroups but i couldn't proceed in the ...