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 ...

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16 views

Thompson sporadic group presentation and conjugacy class representatives

So if you look here: http://web.mat.bham.ac.uk/atlas/v2.0/spor/Th/ they provide matrices, $a$ and $b$, which generate the Thompson sporadic group. They also give a representative for each conjugacy ...
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2answers
49 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
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2answers
39 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
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2answers
88 views

How did we classify the finite simple groups when we haven't classified the primes?

Why was the classification of the finite non-abelian simple groups "easier" (!!) than the classification of the finite abelian simple groups [the prime numbers], which still doesn't exist? (Does it? ...
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2answers
50 views

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

Any simple group of order 60 is isomorphic to $A_5$ Let $G$ be a simple group of order $60$ .(Assumption) $G$ has a subgroup of order 12 say $H$ . Then by Extended Cayley's Theorem $\exists ...
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43 views

$G$ is a simple group of order $60$.Then $G$ contains a subgroup of order 12

$G$ is a simple group of order $60$. Then show that $G$ contains a subgroup of order 12. * MY TRY: Suppose $G$ has no subgroup of order 12. Let $n_5$ denote the number of Sylow 5-subgroups of $G$. ...
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1answer
120 views

Where do the enormous simple groups come from?

I mean, these simple groups of big order such as 808017424794512875886459904961710757005754368000000000 I think it's order is something similar to a factorial for all those 0s... but I'd like to know ...
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21 views

A question about finite simple groups of Lie type

I have a question about finite simple groups of Lie type. By the classification theorem of finite simple groups, we know that there are four important class of finite simple groups as follows: ...
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2answers
30 views

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j?

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j ? If I denote with $n_i$ the number of i-Sylow subgroups For example if $|G|=180=2^2\cdot3^2\cdot 5$ then if I assume that ...
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2answers
93 views

Direct product of simple groups

Let $G=H_1\times H_2$, $H_1,H_2$ are simple groups. Let $L\vartriangleleft G$ ($L$ isn't trivial). Show that $L$ isomorphic to $H_1$ or $H_2$. I tried to construct "projections" of $L$ on $H_1,H_2$, ...
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93 views

An example of a simple infinite $2$-group

Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian. Take the subgroup ...
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1answer
57 views

Find number of Sylow $3$ and Sylow $5$ subgroups of a simple Group, $G$ of order $60$

So I wanted to check if what I did was correct. I'm not sure if it is and if so what would the correct way to go about this be? So firstly $|G| = 2^2*3*5$. This confirms there is Sylow 3 - ...
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1answer
65 views

What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

So I get the answer as follows (which is correct I believe): {$0$} x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x <$4$> $\vartriangleleft$ $\mathbb Z_7$ x ...
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1answer
25 views

Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or ...
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97 views

No group of order $400$ is simple - clarification

I was reading through a proof that no group of order $400$ is simple which can be found here: http://math.stackexchange.com/a/79644/169389 Here is an outline for a solution. First of all, ...
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2answers
55 views

There is no simple group of order $144$

There is no simple group of order $144$ I have a question to the proof of the statement above (from the book J. Gallian, Contemporary abstract algebra), it is about the index theorem, so I give ...
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0answers
42 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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1answer
73 views

Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
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3answers
260 views

What is a Simple Group?

I am working on this problem from class note: Let $G'$ be a group and let $\phi$ be a homomorphism from $G$ to $G'$. Assume that $G$ is simple, that $|G| \neq 2$, and that $G'$ has a normal ...
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1answer
43 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
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1answer
92 views

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple?

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups ...
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1answer
60 views

Direct product of simple non-abelian groups

Let $G$ be a group, and let $K$ be a normal subgroup of $G$ which is a direct product of simple non-abelian groups. I wanted to prove that $K=C_K(H)[K,\, H]$ for every subgroup $H$ of $G$. Is this ...
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0answers
36 views

Quotient by the intersection of maximal subgroups in a free group.

My question is (a modification of) the following: Let $N=\cap M$, where $M$ is maximal normal of finite index in the free group $G=F_2$. Then what is the group $G/N$? Currently written, this is ...
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2answers
54 views

Infinite Suzuki Groups

Often I found myself on a symbol like $Sz(F)$ where $F$ is an infinite field. What is the definition of an infinite Suzuki group? Are they linear groups? Where I could find some informations about ...
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0answers
66 views

Powers of simple groups

I have heard about the following result: for each natural number $r\ge 2$ and each finite simple non-abelian group $S$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but ...
2
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1answer
75 views

finite simple groups and free groups

It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? ...
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1answer
50 views

The structure of maximal tori in finite simple groups

Let $\mathbf{G}$ be a linear algebraic group over an algebraic closed field of characteristic $p$ and $F$ a proper frobenius map on it with fixed point group $\mathbf{G}^f=G$ such that ...
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1answer
91 views

If G is abelian and simple ,then G is cyclic

True /False .IF G is abelian and simple ,then G is cyclic Solution True If G is an abelian simple group then G is isomorphic to Zp for some prime p
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1answer
117 views

Generalization of a property Of $A_5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_5$ and $\langle H,K\rangle < A_5$. We can show that there exists a maximal subgroup $M$ of $A_5$ such that ...
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1answer
70 views

Simple group with Klein four Sylow

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup ...
5
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2answers
119 views

Conjugacy classes of PSL(6,7)

I need the conjugacy class sizes of projective special linear group PSL(6,7). I couldn't find it by using GAP. Could someone find it?
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1answer
44 views

Is Z2 is simple.

Is Z2 simple.Does every subgroup of Z2 simple. Solution. Z2 is simple, however subgroup of order 1 in Z2 is not simple.
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1answer
50 views

Equivalence classes of (2,3)-pairs in PSL(2,q)

Let $G = PSL(2,q)$. I am interested in the different ways of writing $G$ as an image of the modular group. If $A, B \in G$ are elements of order $2, 3$ respectively, then $\langle A,B \rangle$ is an ...
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1answer
92 views

The Diagonal Subgroup of $A \times A$ is Maximal iff $A$ is Simple

Let $A$ be a group and $G = A \times A$. Define $D= \{(a,a,)\mid a \in A\}$ (the diagonal subgroup of $G$). Prove that $D$ is a maximal subgroup of $G$ if and only if $A$ is simple, i.e. it has no ...
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13 views

A refernce about Cartan matrix

There exist an approach to "Cartan Matrix" in Carter's book "Finite groups of Lie type, conjugacy classes an complex characters" p.23, which seems be different to other definitions of Cartan matrix I ...
3
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1answer
88 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
2
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1answer
122 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
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0answers
16 views

A question about relation between a finite simple group and a linear algebraic group

let $G$ be a linear algebraic group of type $\mathcal{X}$ over an algebraic closed field of characteristic $p$, $K$. suppose $F:G\rightarrow G$ is a Frobenius map and $\mathcal{G}=G^F$ is a finite ...
2
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1answer
66 views

Elements of $\operatorname{Aut}(\operatorname{Aut}(G))$ acting as an identity on $\operatorname{Inn}(G)$, for $G$ a nonabelian, simple group

Let $f$ be an element of $\operatorname{Aut}(\operatorname{Aut} G)$ s.t $f(r)$ is equal to $r$ for all $r \in \operatorname{Inn}(G)$. Prove that $f(j)$ is equal to $j$, $\forall ...
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0answers
34 views

finite simple group of Lie type

let $G$ be a finite simple group of Lie type. would someone please help me! I want to know, is the following expression true or not? " There exists a simple linear algebraic $\mathcal{G}$ over an ...
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2answers
153 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
2
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1answer
99 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
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1answer
87 views

Simple groups with cyclic odd Sylow subgroups

I know that every odd Sylow subgroups of $PSL(2,p)$ is cyclic. Is there any other simple group with cyclic odd Sylow subgroups. Thank you in advance
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35 views

Some things about maximal tori

Let $G$ be a linear algebraic group over an algebraic closed field of characteristic $p\neq 2$. Suppose $\overline{G}_{\sigma}={G}_{\sigma}/Z({G}_{\sigma})$ where ${G}_{\sigma}$ is the set of fixed ...
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4answers
278 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
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0answers
81 views

Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
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0answers
29 views

Using a theorem to find the center of a $p$-sylow subgroup of simple group

I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown ...
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1answer
68 views

The center of Sylow $p$-subgroups of a finite simple group of Lie type

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of ...
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2answers
157 views

Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
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1answer
46 views

regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...