Use with the (group-theory) tag. A group is simple if it has no proper, non-trivial normal subgroups. Equivalently, 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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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 ...
2
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0answers
101 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
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1answer
39 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
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1answer
101 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
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5answers
68 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.
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5answers
162 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 ...
6
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0answers
95 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
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2answers
59 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
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1answer
57 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 ...
29
votes
2answers
378 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:
...
25
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3answers
312 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 ...
9
votes
1answer
114 views
Simple groups of order 168
How would I prove that there is at most one simple group of order 168?
I've already seen that $GL_3(2)$ and $PSL_2(7)$ are simple groups of order 168, and I have seen direct proofs that they are ...
4
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1answer
76 views
On Group of order 30 and 60 .
in this question on yahoo answers ,
http://uk.answers.yahoo.com/question/index?qid=20090823193007AAKQvc2
the answer says ,
" with t = 6, then there are 6 * (5 - 1) = 24 elements of order 5 "
my ...
3
votes
1answer
68 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 $d$.
?
11
votes
1answer
235 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
150 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 $A_6$?
7
votes
1answer
302 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
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3answers
106 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 ...
1
vote
1answer
91 views
Finite simple group with subgroups of same order
Let D be a finite simple group, such that H < D and K < D. Also [D:H]=q and [D:K]=p, where p,q are primes. Want to show that p=q.
I want to come up with a contradiction with one of the ...
2
votes
0answers
30 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
1answer
267 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, ...
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0answers
44 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?
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1answer
89 views
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 ...
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1answer
55 views
simplicity of G
I took an exam today. If I remember correctly question was like this:
let G be a group. if it has "a" element which has exact two conjugates, then G cant be simple.
I answered:
let that two ...
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votes
2answers
237 views
Prove that there are no simple groups of order 224.
Prove that there are no simple groups of order 224.
Let $G$ be a finite group such that $\vert G \vert = 224 = 2^5 \cdot 7$. We know that $n_2 \mid 7$ and $n_2 \equiv 1 \pmod 2$ and we know that $n_7 ...
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1answer
111 views
Proper subgroup of simple groups
Not sure how to do this:
Fix integer n>1. Prove there exist only finitely many simple groups containing proper subgroups of index smaller than or equal to n.
4
votes
1answer
49 views
The extension of $PSL_2(q)$ by its outer automorphism group
Let $q=p^f$ be a prime power. Is $P\Gamma L_2(q)$, the automorphism group of $PSL_2(q)$, a semidirect product of $PSL_2(q)$ by its outer automorphism group $Z_{\gcd(2,q-1)}\times Z_f$? If it is not in ...
6
votes
2answers
231 views
Every normal subgroup of a finite group is contained in some composition series
In this context composition series means the same thing as defined here.
As the title says given a finite group $G$ and $H \unlhd G$ I would like to show there is a composition series containing $H.$
...
6
votes
2answers
236 views
Simple group of order $660$ is isomorphic to a subgroup of $A_{12}$
Prove that the simple group of order $660$ is isomorphic to a subgroup of the alternating group of degree $12$.
I have managed to show that it must be isomorphic to a subgroup of $S_{12}$ ...
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1answer
55 views
Number of prime divisors of the order of $E_8(q)$.
I am trying to compute the number of prime divisors of the order of $E_8(q)$. I am interested in the general solution, but in particular, my problem calls for $q=p^{15}$ (for prime $p$) and $q\equiv ...
4
votes
3answers
128 views
Proof that every element of A_5 is an involution or a product of two involutions?
It can be verified with brute force that the alternating group on 5 elements ($A_5$) has the property that every member is either an involution or can be written as the product of two involutions. Is ...
10
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2answers
300 views
Is there an infinite simple group with no element of order $2$?
According to the Feit-Thompson theorem, every group of odd order is solvable and thus every finite nonabelian simple group has even order. Thus every finite nonabelian simple group has an involution ...
6
votes
1answer
226 views
Presentations for alternating groups
Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators
$a_n=(123),b_n=(1,2,\ldots,n)$?
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1answer
194 views
There is no simple group of order $448=2^6\cdot 7$
How can I prove that there is no simple group of order $448=2^6\cdot 7$? I tried with Sylow's theorems, I proved that (if $G$ is simple) the number of 2-Sylows is 7 and that the number of 7-Sylows is ...
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1answer
404 views
Simple group of order 168
Let $G$ be a simple group of order 168, I have to show that it has at least 14 elements of order 3.
Using Sylow's theorems I proved that if $n_3$ is the number of 3-Sylows then $n_3\in\{4,7,28\}$, ...
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2answers
156 views
Coincidences with orders of simple groups
The projective special linear groups $PSL(2,4)$, $PSL(2,5)$ and $PSL(2,9)$ have the property that their orders equal the order of an alternating group. They are also isomorphic to the respective ...
5
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2answers
154 views
Infinite 2-generated simple group
I am looking for a concrete example of an infinite simple group with two generators. Ideally, one generator has order 2, the other 3, but if there is a nice example without this requirement, it will ...
3
votes
1answer
62 views
Number counting functions related to simple groups and asymptotic law of distribution
We say that an positive integer $n$ is a
simple number if there exist a non abelian simple group of order $n$. Denote by $\mathfrak{s}$ this set.
prime-power number if it is of the form $n=p^a$, ...
3
votes
1answer
93 views
Infinitely many simple groups with conditions on order?
If $G$ is a non abelian finite simple group, we know that $4$ divides $|G|$.
More precisely there are infinitely many finite simple groups $G$ such that $v_2(|G|)=2$, just consider ...
6
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1answer
251 views
What makes simple groups so special?
The classification of finite simple groups was one of the most important problem in group theory. But what makes simple groups so interesting and special?
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1answer
433 views
$A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ only simple group of order 360
How can one show, without the use of character theory, that $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ is, up to isomorphism, the only simple group of order 360?
7
votes
2answers
310 views
Why is $PGL(2,4)$ isomorphic to $A_5$
In the tradition of this question,
why is $\operatorname{PGL}(2,4)\cong A_5$?
1
vote
2answers
310 views
$M$ is a maximal normal subgroup iff $G/M$ is simple
Fraleigh(7th) Thorem15.18. $M$ is a maximal normal subgroup of G if and only if $G/M$ is simple.
I have a problem in "if" part. To prove ($\Leftarrow$) direction, assume that $N$ is a normal ...
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votes
1answer
106 views
Proving that a simple group $G$ with $H <G$ and $|G:H|=n$ is isomorphic to a subgroup of $S_n$
I'm trying to prove that if $G$ is a simple group and $H$ is a proper subgroup of $G$ of index $n$ then $G$ is isomorphic to a subgroup of $S_n$.
To do this, I've been trying to find a homomorphism ...
5
votes
1answer
147 views
Internal Structure of A_7
In $A_7$,
1) Are all subgroups of order 168 are conjugate? ($A_7$ contains a simple group of order 168).
2)Does it contain an abelian group of order 12? What is the largest order of abelian group?
...
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1answer
174 views
5
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0answers
343 views
Alternating and special orthogonal groups which are simple
I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if ...
5
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0answers
139 views
Dual modules and first cohomology
Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module.
What hypotheses are needed on $G$, $M$ to ...
2
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1answer
191 views
questions about simple groups
how to show that there is no simple group of order $1755 = 3^3 \cdot 5 \cdot 13$? Thank you very much.
5
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2answers
407 views
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
I want to show that there are no simple groups of order $p^{k}(p+1)$ where $k>0$ and $p$ is a prime number.
So suppose there is such a group. Then if we let $n_{p}$ denote the number of $p$-Sylow ...
