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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simple groups of lie type

Let G be a finite simple group of lie type. I want to know is G a finite group of lie type? Also if yes, when type of G is X as a finite simple group of lie type, has G same type as a finite group of ...
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2answers
201 views

A group homomorphism from a simple group is injective

Let $G_1$ be a simple group, that is the only normal subgroups of $G_1$ are itself and the trivial subgroup. If $\phi : G_1 \rightarrow G_2$ is a group homomorphism, does that mean $\phi$ is ...
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1answer
72 views

If $|G|= 2^{3}3^{3}11$, then $G$ is not simple

I have this problem in my notes: If $|G|=2^{3}3^{3}11$, then $G$ is not simple The instructor solved it in a way that I could not follow. The solution I have is attached below. If someone could ...
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2answers
54 views

for prime $q\ge p$, integer $m\ge 0$, any group with order $p^2 q^m$ is not simple

I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup. Is there ...
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21 views

Nodes of the Dynkin diagram for even-dimensional orthogonal groups

I'm reading Wilson's book "The Finite Simple Groups", specifically the sections on the orthogonal groups. In section 3.7.4, he discusses subgroups of the orthogonal groups. The Dynkin diagram for the ...
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1answer
74 views

Why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$? [closed]

Can anyone please clarify me on why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$ for some positive constant $k$? Thanks
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1answer
26 views

Showing group of order 9555 is not simple by using following Hint

There is already another proof of this theorem, but I'm curious about solving this problem as written in Dummit&Foote(Ex.6.2 12) Show there is no simple group $G$ with $|G|=9555$. No simple ...
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1answer
57 views

If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple

I'm looking for verification: My claim: If $G$ is a finite group and $H$ is a (proper)subgroup of index $k>1$, where $k! \leq |G|$, then $G$ is not simple. Proof: Consider the set of left cosets ...
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1answer
25 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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65 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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53 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
99 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
56 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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48 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
126 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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24 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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31 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 ...
2
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2answers
117 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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1answer
105 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
71 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
66 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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2answers
105 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
60 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 ...
3
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44 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
77 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
267 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
50 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
108 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
70 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
44 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
60 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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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
76 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
51 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
123 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
2
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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 ...
3
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1answer
72 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
122 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
66 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
51 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 ...
2
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1answer
104 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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16 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
107 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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0answers
54 views

Sufficiently transitive implies alternating sans Enormous Theorem

According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis ...
2
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1answer
135 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 ...
2
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1answer
69 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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2answers
172 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
95 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