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.

A group is simple if it has no proper, non-trivial normal subgroups (a subgroup $H\leq G$ is proper if $G\neq H$, and is non-trivial is $H\neq 1$). 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.

Simple groups can be seen as the "building blocks" of groups. This is explained in this question.

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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 http://www.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html Because it has order 60 and two…
user58512
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Why are there only a finite number of sporadic simple groups?

Is there any overarching reason why, after excluding the infinite classes of finite simple groups (cyclic, alternating, Lie-type), what remains---the sporadic, exceptional finite simple groups, is in fact a finite list (just 26)? In some sense, the…
Joseph O'Rourke
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Ignoring elements of small order in the simple group of order $60$

The simple group of order $60$ can be generated by the permutations $(1,2)(3,4)$ and $(1,3,5)$, but all you need to do is square the first one and it becomes the identity. Can't we find a version of the simple group where the elements of small…
Jack Schmidt
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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: 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71 starts with a very high power of 2, then the powers decrease and…
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Distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem. Why?

It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number of Sylow $p$-subgroups. It is explained that the…
Jonathan Beardsley
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No simple group of order $300$

So I've been trying to prove that there's no simple group of order $300$. This is what I did and I was wondering if it was enough. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there would be $6$ Sylow $5$-subgroups, one of which will…
Nana
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Proof that all abelian simple groups are cyclic groups of prime order

Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks! Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| > 1$, so for some $x\in G$ we have…
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Why is the Monster group the largest sporadic finite simple 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 bigger group out there? Beyond that long proof, is…
riemannium
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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 $f_h(g)=hgh^{-1}, \forall g\in G$, where $h\in G$ . Now if $G_1$ is a…
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$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?
Nathan Portland
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Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?

This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that: $L_4(2)$ and $L_3(4)$ both have order $20160$ $O_{2n+1}(q)$ and $S_{2n}(q)$ have the same order for $q$ odd, $n > 2$ I think…
Simon Nickerson
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The "architecture" of a finite group

I think that the aim of finite group theory is the following: Given an arbitrary finite group $G$, study completely the subgroup structure of $G$. There are at least two ways to achieve this purpose: 1) The approach with simple groups. Thanks to…
Dubious
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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 isomorphic to $C_2 \times \mathrm{PSL}(2,q)$. At…
Dune
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Why $PSL_3(\mathbb F_2)\cong PSL_2(\mathbb F_7)$?

Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic? Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, maybe?
Grigory M
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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 equal. Now I'd like to show there is only one simple…
user58512
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