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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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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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.
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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 ...
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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?
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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 ...
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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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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 ...
