5
votes
1answer
52 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
vote
1answer
50 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
32 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
56 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
46 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 $[G:K]=[K:H]$ ? ...
5
votes
2answers
72 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 ...
1
vote
1answer
56 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
31 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
3
votes
1answer
68 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
66 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
1answer
85 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
40 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 ...
2
votes
2answers
36 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
2answers
98 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
145 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 ...
4
votes
1answer
124 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
72 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 ...
-1
votes
2answers
70 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
78 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: ...
1
vote
1answer
68 views

Homomorphism defined on a simple group

Can anybody please help me with this question??? Let $G$ and $G'$ be groups with $G$ simple. Prove that if $|G| > |G'|$, then the only homomorphism $\varphi : G → G'$ is the trivial one. ...
7
votes
0answers
88 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
2
votes
0answers
40 views

Permutation representation argument validity

I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you. Problem statement: Let $G$ be a group of order 150. Let $H$ be a ...
0
votes
1answer
76 views

Simple group with order $\geq n!$ cannot have subgroup of index $n$.

My problem is as seen in the title: For positive integer $n>1$, prove that a simple group with order $\geq n!$ cannot have subgroup of index $n$. Could anyone give me some hints on how to ...
2
votes
2answers
66 views

If $G_1 \leq G_2 \leq \cdots \leq G$ such that $G = \bigcup_{i=1}^\infty G_i$ and each $G_i$ is simple then $G$ is simple.

Prove that if there exists a chain of subgroups $G_1 \leq G_2 \leq \cdots \leq G$ such that $G = \bigcup_{i=1}^\infty G_i$ and each $G_i$ is simple then $G$ is simple. I have proved the following ...
8
votes
2answers
165 views

Verifying finite simple groups

The classification of finite simple groups required thousands of pages in journals. The end result is that a finite group is simple if and only if it is on a list of 26 sporadic groups and several ...
3
votes
2answers
105 views

Who did first use the term “simple” in group sense and why?

Who did first use the term "simple" in group sense? More appreciated if I learn the original word(might be in French, Galois???)... and also... Why do you think that this term is chosen ...
3
votes
1answer
52 views

Are quasinilpotent groups a Fitting class?

A finite group is called quasinilpotent if it induces inner automorphisms on all of its chief factors. A solvable group is quasinilpotent iff it centralizes all of its (necessarily abelian) chief ...
7
votes
1answer
136 views

Are there any distinct finite simple groups with the same order?

In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two ...
11
votes
1answer
607 views

Is there a simple group of any (infinite) size?

I'm trying to show that for any infinite cardinal $\kappa$ there is a simple group $G$ of size $\kappa$, I tried to use the compactness theorem and then ascending Löwenheim-Skolem, but this is ...
7
votes
1answer
63 views

Are the quotient groups in a composition sequence necessarily subgroups?

Does there exist a finite group G and a normal subgroup N of G so that G/N is a simple group and G/N is not isomorphic to any subgroup of G ?
2
votes
2answers
109 views

Simple groups some interesting properties

I have found some interesting results as follows: If $o(G)\not|\ i(H)!$ then $H$ contains a non-trivial normal subgroup of $G$, where $i(H)=[G:H]$. 2.If $o(G)=2m$, where m is an odd prime number ...
3
votes
0answers
141 views

The Monster group

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
0
votes
1answer
59 views

Projective linear group - solvable

Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
4
votes
1answer
156 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
2
votes
5answers
246 views

Group of order 9 is simple

Is't true that a group of order $9$ is simple? How can it be proved or disproved.
6
votes
5answers
405 views

Definition of Simple Group

Herstein defined the definition of a simple group as follows: A group is said to be simple if it has no non-trivial homomorphic image. Please help me to understand what is meant by non-trivial ...
10
votes
0answers
180 views

How confident can we be about the validity of the classification of finite simple groups?

The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed. How certain can we be that The proof of ...
1
vote
2answers
105 views

Examples of profinite simple groups

The standard example of an infinite simple group is $A_\infty$, the direct limit of the alternating groups under the obvious injections. Are there also examples of infinite simple groups arising as ...
4
votes
1answer
68 views

Sylow $p$-subgroups of $SO_3(\mathbb{F}_p)$

Let $G=SO_3(\mathbb{F}_p)$. We have $|G|=p(p^2-1)$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Is it true that $n_p=p+1$? We have the two conditions $n_p\equiv 1\mod p$ $n_p\mid ...
30
votes
2answers
443 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
28
votes
3answers
532 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
4
votes
1answer
167 views

On Group of order $30$ and $60$.

In this question on yahoo answers , the answer says , "with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ " my question is , how did " 6 * ( 5 - 1 ) " come from ? Which ...
4
votes
1answer
86 views

Is there a simple and a non-simple group with same numbers of elements of each order.

Are there finite groups $G$ and $H$ such that: $n:=|G|=|H|$. $G$ is simple. $H$ is not simple. for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ?
14
votes
1answer
315 views

The “architecture” of a finite group

I think that the aim of the finite group theory is the following: Given a generic finite group $G$, study completely the subgroup structure of $G$. There are at least two ways to achieve this ...
3
votes
2answers
285 views

Non-Abelian simple group of order $120$

Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
8
votes
1answer
337 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
6
votes
3answers
195 views

Every simple group of odd order is isomorphic to $\mathbb{Z}_{p} $ iff every group of odd order is solvable

I'm trying to prove the following claims are equivalent: Every simple group of odd order is of the type $\mathbb{Z}_{p}$ for prime $p$ Every group of odd order is solvable. Getting from 2 to 1 was ...
2
votes
0answers
41 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
2
votes
2answers
702 views

Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple

Here's a question I've been asked; Given distinct primes $p,q,r$, show that any group $G$ of order $pqr$ is not simple. So far, my idea has been to individually check each possible proper subgroup, ...
0
votes
0answers
62 views

Soluble subgroups of finite classical simple groups

What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?