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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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Is there a simple abelian group $G$ with infinite order?

I am reading "An Introduction to Algebraic Systems" by Kazuo Matsuzaka. There is the following problem in this book: On p.80 Problem 8: Show that a simple abelian group $G \neq \{e\}$ is a ...
5
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1answer
34 views

Relation between the order of an element of a group and their character in a simple group

Let $\chi$ be the representation of a finite group $G$. Let $g \in G$ be an element of order 2. If $G$ is a simple group but not cyclic of order 2, prove that $\chi(g) \equiv \chi(1) \mod 4$. Proof ...
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30 views

When does a simple Lie group contain a nonsimple subgroup?

The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $\text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well. The conformal group of ...
21
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1answer
202 views

On automorphisms of groups which extend as automorphisms to every larger group

For a group $G$, let $\operatorname{Aut}(G)$ denote the group of all automorphisms of $G$ and $\operatorname{Inn}(G)$ denote the subgroup of all autmorphisms which is of the form $f_h(g)=hgh^{-1}, \...
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62 views

No group of order 10000 is simple

A proof of this fact was already given here: No group of order 10,000 is simple However, I am wondering whether or not the following proof works as well: By way of contradiction, suppose $G$ is ...
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29 views

The cases in proving that a group of order 90 is not simple

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a ...
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42 views

Monster group poster

I have seen a number of questions here on how to intuitively understand the Monster Group. My questions is, is there a way one can create an image or series of images suitable for putting on a poster ...
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1answer
43 views

If all homomorphisms $f:G→H$ are trivial or injective, then G is simple.

Let $G$ be a nontrivial group. Show that $G$ is simple if and only if, for every group $H$ and homomorphism $f:G→H$, either $f$ is trivial or $f$ is injective. So I have already proved that if $f$ ...
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1answer
32 views

Minimum size of index of a proper subgroup of a finite, non-abelian simple group $G$

Let $G$ be a finite non-abelian simple group and $p$ the largest prime divisor of $|G|$. Show that if $H < G $ then $|G : H | \geq p $. This is from a chapter of a book about group actions, ...
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1answer
37 views

Show a group cannot have order $2n$ for some odd $n\gt 1$ and be simple (without Cauchy's Theorem) [duplicate]

We were asked to prove the following today: Let $G$ be a group. Prove that if $G$ has order $2n$ for some odd integer $n$ greater than $1$, then $G$ contains a proper non-trivial normal subgroup ...
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16 views

A central extension of a simple group

Let $p$ be a prime and $i$ be a positive integer. Is there a finite group $G$ such that $G'=G$, $G/Z(G)$ is simple non-abelian, $Z(G)=p^i$ and having a central automorphism not fixing element-wise the ...
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2answers
130 views

Show that no group of order 48 is simple

Show that no group of order 48 is simple I was wondering if I was allowed to do something along this line of thinking: Let $n_2$ be the number of $2$-Sylow groups. $n_2$ is limited to $1$ and $3$ ...
4
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1answer
33 views

If $G$ is a simple group of order $60$, how can we there exists some $G\to A_6$ not trivial without using the fact that $G\cong A_5$?

If $G$ is simple, any homomorphism out of $G$ must be trivial or injective since the kernel of a homomorphism is a normal subgroup. If $G$ is simple of order $60$, how can I show that there exists an ...
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53 views

What is the connection between simple groups, composition series, and solvable groups?

I am reading Dummit and Foote section 3.4 (Composition series and the Holder program) and I am having trouble understanding how these concepts are connected. A group $G$ is simple if the only normal ...
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1answer
52 views

When a simple group normalizes a subnormal subgroup of a finite group

Out of curiosity I am working my way through Isaacs's Finite Group Theory and am stuck on problem 2A.4: Let $(G,*)$ be a finite group with simple subgroup $N$ such that $\forall H \lhd\lhd G: ...
5
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1answer
127 views

How to show that Group of order $2376$ is not simple

How to show that Group of order $2376$ is not simple, Now I know that $2376=2^3.3^3.11$ So, $n_{11}=1,12$(Are there any other possibilities? According to my calculation; these are the all) Now if I ...
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2answers
54 views

A group of order $p^nm$ isn't simple

Suppose $|G|=p^nm$ with $1<m<p$. Show that $G$ is not simple. I know it has to do with group actions. My idea is to consider a subgroup of order $p^n$, call it $H$. It has index $m$, so there ...
4
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1answer
92 views

For odd primes $p$, are finite groups with self-normalizing Sylow $p$-subgroups solvable?

Is it the case that for odd primes $p\geq5$, all finite groups with self-normalizing Sylow $p$-subgroups are solvable? The simple group of order 168 shows that this conjecture does not hold for $p=2$. ...
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1answer
111 views

Can one prove that a group of order 150 is not simple using element counting?

I'd like to know whether it's possible to show that a group of order 150 is not simple using only element counting (or mostly element counting). I have seen a solution to this problem here (Every ...
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1answer
72 views

A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple. [duplicate]

Let $G$ be a finite group of even order, whose $2$-Sylow subgroups are cyclic. Show that $G$ is not simple. I was trying to use Cayley theoem, place $G$ in some $S_m$ and get a contradiction with $...
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2answers
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Simple groups in group theory 2

Problem Let $G$ be a finite group and $H \subset G$ be a subgroup of index $\lvert G:H \rvert =n$. (a) Show that $\lvert H:(H\cap gHg^{-1})\rvert \leq n$ for all $g\in G$ (b) If $H$ is a ...
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Could every finite simple group be related to a pair of Lie Groups?

In terms of the classification of simple groups, it is known that every finite simple group is either: Cyclic Alternating Of Lie type One of the 26 Sporadic groups On the other hand there are 5 ...
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45 views

Showing that a group of order 96 is not simple. [duplicate]

I am trying to prove that a group $G$ of order 96 is not simple. My approach is the following: Let $P$ be a Sylow 3 subgroup and let $Q$ be a Sylow 2 subgroup. Then by the Sylow's theorems we know ...
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1answer
139 views

Prove that there is no simple group of order $144$

I was reading the following proof for that question (Joanpemos' answer)- How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections. And I understood it well up to ...
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2answers
83 views

The order of the group $A_n$

I was reading the following proof of why there is no simple group of order $120$: A group of order $120$ cannot be simple And I couldn't understand the following: "so $A_6$ has a subgroup of order $...
2
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3answers
158 views

Show that every group of order $5865$ is cyclic

I need help in proving that every group of order $5865$ is cyclic. I thought at the beginning that if I show that every such group is simple and abelian, It will be isomorphic to some $\mathbb{Z}/p\...
2
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1answer
56 views

A finite simple group with subgroup for every divisor of $|G|$ is abelian

Let $G$ be a finite simple group. Assume that for every positive integer $d$ that divides $|G$, there is a subgroup $H$ of $G$ such that $|H| = d$. Prove that $G$ is abelian. So I'm really out of ...
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1answer
71 views

How to prove that there is no simple group of order 10 000 000?

How to prove that there is no simple group of order 10 000 000? Here is what I have so far: Let $G$ be such a group, we have $|G|=10 000 000=2^7*5^7$. To prove that $G$ is not simple a solution ...
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2answers
69 views

How to show that the intersection of 2 distinct p-Sylows (p prime) is the identity?

Let us suppose that $G$ is a simple group of order 90. Show that 2 distinct 3-Sylows cannot contain the same element $g \neq e_G$ where $e_G$ is the identity. First we can compute the number of 3-...
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3answers
136 views

How to prove that a finite group of order 280 is not simple?

Let $G$ be a finite group of order 280. How to prove that $G$ is not simple? A way to do it is to prove that there exists a p-Sylow subgroup of G that is normal, ie that there is a unique p-Sylow ...
2
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1answer
47 views

Prove a group of order $p(p+q)$ is not simple for primes $p$ and $q$ where $ p \geq q$.

So my question is the title; I have been self studying for an exam coming up and this problem showed up. I am not sure how to proceed I can't seem to see how the $n_p(G)$ is related to $p+q$. Any hint ...
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0answers
51 views

Minimal degree representation of finite simple groups

For this question I am taking a representation of a group $G$ to be a homomorphism from $G$ to $GL_n(K)$ for some field $K$. The degree of the representation is $n$. I am trying to understand the ...
2
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1answer
56 views

Groups for which all normal subgroups are perfect

I am trying to understand the following property of groups. A group $G$ is perfect if $G=[G,G]$. So call $G$ extra-perfect if every normal subgroup of $G$ is perfect. One obvious class of extra-...
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38 views

Conjugacy class with $1$ element in an Infinite Simple Group

We would like to show that if $G$ is an infinite simple group, then the only conjugacy class of exactly one element is $\{1_G\}$. My thoughts: We want to proove that if $|\mathrm{orb}(x)|=1\iff \...
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1answer
67 views

Is non-abelian simple group complete?

Let $G$ be a non-abelian simple group. I wonder if $G$ is complete; i.e., $\mathrm{Inn}\,G = \mathrm{Aut}\,G$. Although I am an elementary learner, I know, just by simple calculation, that $\mathrm{...
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1answer
32 views

Proving that in a finite abelian group, there exist a chain of subgroups so that the quotient is simple.

If $G$ is an abelian finite group, how can I prove that there exist a chain of subgroups $$\{1\}\subseteq H_1 \subseteq H_2 \subseteq \ldots \subseteq H_n=G$$ So that $H_{i+1}/H_i$ is simple? I don't ...
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2answers
28 views

Why do the integers modulo a composite positive number never form a simple group under addition?

Let $p,q \in \mathbf{Z}$ such that $p > 1$ and $q > 1$. Then $\mathbf{Z}/pq\mathbf{Z}$ does not form a simple group under addition. Why?
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1answer
84 views

Coprime action of simple groups

Let $A$ act on $G$ coprimely by automorphism where $G$ is a nonabelian simple group. Does it imply that $|A|=p$? (where $p$ is a prime number) If not, Is there any source that examines such cases ...
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78 views

Are all non-abelian groups not simple?

We know that for all $n\geq 3$ the group $S_n$ is non-abelian. We also know that for all $n\geq 3,$ the group $S_n$ is not simple, because $A_n$ is a normal subgroup of $S_n$ which is not trivial. ...
4
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1answer
605 views

Any simple group of order $60$ is isomorphic to $A_5$

There are many famous proofs. However, I cannot find a solution using the following method which is also clear. It’s absolutely correct. Has anyone seen this proof before, and do you have a reference? ...
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364 views

Groups of order $180$, $540$, $1080$ are not simple.

Here's how I solve the problems. Thanks for pointing out what might be the weakness of my solutions. Actually, what I want are other ways of solving this kind of problems, appart from counting the ...
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86 views

Isomorphy of simple groups of order 360 : a proof with a presentation

It is well known that all simple groups of order 360 are isomorphic with the alternating group $A_{6}$. Cole's original proof is here on StackExchange : $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ ...
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1answer
46 views

If $G$ is non-abelian and simple then $|G|$ divides $n_p!/2$

If $G$ is non-abelian and simple then $|G|$ divides $n_p!/2$ where $n_p$ is the number of Sylow p-subgroups. I am struggling to prove this. I am considering the conjugation action so I know $G$ must ...
4
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0answers
92 views

Finite groups with 15 or 16 conjugacy classes [closed]

How can I classify all almost simple groups with 15 or 16 conjugacy classes? A finite group $G$ is almost simple if there is a non-abelian simple group $S$ such that $S\trianglelefteq G\leq \...
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1answer
215 views

Center of Simple Abelian Group and Simple Nonabelian Group

I just read about short topic about simple group and I found problem about center of simple abelian group and nonabelian From definition of simple, it must have no proper non trivial normal subgroup....
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1answer
122 views

Every finitely generated group has simple quotient

I've read that every finitely generated group has a simple quotient. Is it obvious?
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1answer
327 views

Discovery of the first Janko Group

Recently, I was reading about Janko's discovery of $J_1$, the first “modern” sporadic simple group. Janko and others were trying to classify all finite simple groups with an involution centralizer ...
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1answer
97 views

Prove that there is no subgroup of index $6$ in a simple group of order $240$

Let $G$ be a group of order $240=2^4\cdot 3\cdot 5$. Assume $G$ is simple, then show that there is no subgroup of index 2, 3, 4 or 5 show that there is no subgroup of index 6 For ...
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82 views

A group of order 189 is not simple

I seek to prove that a group G of order 189 is not simple. So, for contradiction, I assume G is simple. $|G|=189=3^3 7$. Now, by the Sylow theorems, $n(7)=1+7k$ divides $3^3=27$. But this is only ...
3
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2answers
88 views

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple.

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple. By the Sylow theorem, we have that the number of $2$-sylow subgroups of $G$ $n_2$ satisfy that $$ n_2 \equiv1\mod2\mbox{ ...