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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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A finite non-abelian group of order $n$ that for every divisor of $n$ has a subgroup is not simple

Let $G$ be non-abelian group of order $n$. Also, for every $k$ which is a divisor of $n$ , there is a subgroup of $G$ of order $k$. I want to prove that $G$ is not simple. Well, from what is given, I ...
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Why is the number of Sylow 2 subgroups of simple group with order 60 not able to be 1 or 3?

I want to show that a simple group of order 60 is isomorphic to $A_5$. In the process, I am stuck at the part in which I have to show that the number of Sylow 2 subgroups (whose orders are 4) cannot ...
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No simple groups of given order.

I am trying to show the following: Prove that there are no simple groups of the given order: 42. 200. 231. 255. I understand that they need to be broken down into their prime factors. I was ...
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1answer
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Automorphisms of $D_4(q)$ (Chevalley group)

As a follow-up to my previous question, I need to investigate the action of certain outer automorphisms of $D_4(q)$ on specific $2$-subgroups (the defect groups of its blocks, if anyone is interested)....
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1answer
40 views

Structure of the outer automorphism group of $D_n(q)$

From the ATLAS, I know that the outer automorphism group of the Chevalley group $D_n(q)$, $q=p^f$ for some prime $p$ and some $n$ even and $n>4$, is a semidirect product of three groups, $(C_d \...
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1answer
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simplicity of the Janko group $J_1$

In the paper, Janko shows the simplicity of Janko group $J_1$ at the Lemma 2.1. In this proof it says "By a transfer theorem all involutions are conjugate in $G$", but I cannot understand. Some ...
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37 views

Using the number of Sylow 2-subgoups of $G$.

I want to show that a group of $G$ of order $56$ is not simple using the number of Sylow $2$-subgroups of $G$ with $n_{2}=7$ and considering two Sylow $2$-subgroup $P_{1},P_{2}$ of $G$. (I know the ...
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Does there exist a characteristically simple group, which is not a direct product of simple groups?

Does there exist a characteristically simple group, which is not a direct product of simple groups? A characteristically simple group is a group without non-trivial proper characteristic subgroups. ...
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Abelian subgroups of a simple group

Let $G$ be a finite non-abelian simple group, and $A \leqslant G$ be an abelian subgroup. How large $A$ can be? There exists any bound of the type $|A| \leq |G|^r$ for some $r<1$? How can I prove ...
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Does there exist some sort of classification of finite marginally simple groups?

Let’s call a group marginally simple if it does not have any non-trivial marginal subgroup (strict definition of marginal subgroups and brief overview of their properties can be found here: https://...
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1answer
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Proof that if $G$ permutes the factors of $T^k$ transitively, $G$ is maximal in $T^k \rtimes G$

Suppose $T$ is a finite non-abelian simple group, $Inn(T^k) \leq G \leq Aut(T^k)$, and $G$ permutes the factors of $T^k$ transitively. Show that $G$ is a maximal subgroup in $T^k \rtimes G$, (where $G$...
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Let $G$ be a finite simple group and let $H\subset G$ be a abelian subgroup of index $|G:H|=p$. Prove that $H = \{e\}$. [closed]

Let $G$ be a finite simple group and let $H\subset G$ be a abelian subgroup of index $|G:H|=p$ for $p$ some prime. Prove that $H = \{e\}$. I don't know how to start... Plz someone help me.
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$PSL(2,13)$ has no subgroup of prime index.

I want to show that $PSL(2,13)$ has no subgroup of prime index,where $PSL(2,13) = \frac{SL(2,13)}{\brace-I,I}$. We have the below fact. 《If $G$ be a simple group and $H$ be a subgroup of $G$ such ...
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Group of order $108$ not simple using Sylow theorems on Sylow-$2$-subgroups

Show that any group of order $108$ is not simple. I can show this using Sylow theorems on Sylow-$3$ subgroups. I was not able to completely justify for Sylow-$2$ subgroups though. In case of Sylow-$...
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1answer
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Find the number of group homomorphism between $A_5$ and $S_5$.

The above question is based on this answer to a similar question. I just want to apply what has been pointed out in that answer to this question. So we are interested in number of homomorphisms $f:...
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2answers
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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 ...
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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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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 ...
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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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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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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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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
45 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
34 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
39 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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17 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
212 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 $3$ ...
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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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79 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: ...
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1answer
134 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
69 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 ...
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1answer
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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
114 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
74 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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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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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
152 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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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 $...
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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\...
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1answer
70 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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72 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
158 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 ...
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1answer
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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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55 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 ...
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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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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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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{...