Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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212 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
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41 views

Direct products of simple non-abelian groups 2

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ a simple non-abelian subgroup of $G$ such that $Z(G)\leq N$ and $\frac{G}{N}\cong A$ be a non-abelian simple group. Is it true that $G=N\times A$ ...
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32 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
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94 views

What does $aba^{-1}b^{-1} \notin H$ imply?

​I am working on a problem on commutator subgroup of finite group. Long story short, I was given $H < G$ and $H' \neq H$ and am aiming to prove $H \lhd G$. As you know that $H'$ is commutator ...
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95 views

How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: ...
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79 views

Why do Sylow $p$-groups in finite simple group have trivial intersection?

I'm looking for an argument for the fact that two Sylow $p$-subgroups in a finite simple group have trivial intersection. In the case that the order of the Sylow subgroups is the prime $p$ it is easy, ...
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23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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56 views

Kernel and image of a homomorphism $SL(2,5)\to S_5$

Since $SL(2,5)$ has a subgroup of index $5$, I can use the left coset action to define a homomorphism between $SL(2,5)$ and $S_5$. How can I find the kernel and the image of this homomorphism? ...
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30 views

How we can conclude that $p\nmid \sum_{x\in H}|x^G|$ in a group with some elements of order $2p$?

Let $G$ be a finite group such that has some elements of order $2p$, where $p$ is an odd prime. Let $H$ be the set of all elements of order $2p$ in $G$. We can show $G$ acts on $H$ by conjugation. So ...
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47 views

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. $H$ cyclic), and among them, let $(H_{i_1})$ be the sequence of ...
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42 views

Constructing irreducible representations of quaternion group over $\mathbb{Q}$

I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework. My question is about constructing all irreducible representations of ...
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40 views

proof theorem 13.9 on finite permutation groups of Wielandt book

I can't prove this theorem on finite permutation groups of Wielandt book. Theorem 13.9: Let p be a prime and G a primitive group of degree n=p+k with k≥3. if G contains an element of degree and order ...
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35 views

$\mathrm{Aut}(G)$ vs. $\mathrm{Aut}(H)$ where $H$ is a maximal abelian subgroup

Can I find a finite group $G$ and a maximal abelian subgroup $H$ such that $ \mathrm{Aut} (H)$ is not isomorphic to a subgroup of $\mathrm{Aut}(G)$?
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28 views

show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ .

suppose that $G$ is finite group and $p$ is a prime number,then show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ where $q$ is arbitrary prime number and $q\neq p$ . ($O^{p}(G)$ ...
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46 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
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31 views

If $\sigma$ is a cycle of length $r$, then it has order $r$?

I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of ...
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38 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
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20 views

Commutators inside center of factor subgroups

Let $G$ be a finite group. Assume that $K \subseteq L \trianglelefteq G$ with $K \trianglelefteq G$. Then $L /K \subseteq \textbf{Z}(G / K)$ if, and only if $[G,L] \subseteq K$. I know it is related ...
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25 views

Prove that $Q_{8}'=\{1,-1\}$. Is my proof correct?

Prove that Prove that $Q_{8}'=\{1,-1\}$. My proof: $Q_{8}'\neq \{1\}$, because $Q_{8}$ is not abelian. $|Q_{8}/\{-1,1\}|=4$. So $Q_{8}/\{-1,1\} \cong \mathbb{Z}_4$ or $Q_{8}/\{-1,1\} \cong ...
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29 views

Generating a group by its $q$-elements.

Let $G=PQ$ be a solvable group where $P$ and $Q$ are $p$-subgroup and $q$-subgroup of $G$ respectively. Also suppose that $Q$ is not normal in $G$. Is it true that the group generated by all ...
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34 views

Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
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45 views

Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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30 views

If all sylow subgroups are cyclic, prove that G is solvable

I came across a statement which I am unable to prove by myself that if $G$ is a finite group then if all its sylow subgroups are cyclic, prove that G is solvable. If it has been asked before please ...
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36 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
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45 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...
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23 views

Find generators in GF(19)

I have 2 questions. Finding generators in GF(19) is similar to finding generators in GF(2^p)? Is primitive polynomial needed to find generators for GF(19)? Thanks a lot. Ya Ali.
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35 views

On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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23 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
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47 views

Help with proof for problem 14 chapter 5 Dummit and Foote.

For any group $G$ define the dual group of $G$ (denoted $\hat{G}$) to be the set of all homomorphisms from $G$ into the multiplicitive group of roots of unity in $\mathbb{C}$. Define a group operation ...
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55 views

Generalization of a property of finite nilpotent groups

Let $G$ be a finite nilpotent group and $M$ be a maximal subgroup of $G$. If $H$ is a proper non-trivial subgroup of $G$ such that $H\not\leq M$, then we can show that $H\cap M$ is a maximal subgroup ...
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42 views

Lucido's three prime lemma

I am looking for proof of this statement I encountered in a paper. $\textbf{(Lucido’s Three Primes Lemma)}$- Let $G$ be a finite solvable group. If $p, q, r $ are distinct primes dividing |$G$|, ...
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19 views

SO(3,q) to PGL(2,q)

Can anyone suggest a reference to an explicit formula giving, in the standard matrix notation, a homomorphism from SO(3,q) to PGL(2,q) (classical matrix groups: orthogonal of dimension 3 and ...
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28 views

Is membership in the index 2 subgroup of $Sp_4(\mathbb{F}_2)$ detected by a polynomial in the matrix entries?

I learned from Magma that $Sp_4(\mathbb{F}_2)$ has an index-2 subgroup isomorphic to $A_6$. Is it possible, given a matrix $M\in Sp_4(\mathbb{F}_2)$, to detect membership in this subgroup using a ...
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13 views

Characterise all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$

The title pretty much explains everything. Is it possible to give an easy characterisation of all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$?
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32 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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61 views

Can a chain subcomplex be a direct summand but it's 'origin' is not? (group homology)

Let $H\!\leq\!G$ be finite groups and $C_\ast(H)\!\leq\!C_\ast(G)$ their bar complexes (each $C_k(G)$ is a free $R$-module with basis $(G\!\setminus\!\{1\})^k$). Is it possible that $H$ is not a ...
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73 views

Description of an abelian group

I'm again stuck in an algebra exercise. I'm not sure if I understand the problem right. Could it be that I have to show that $\mathbb{Z}[i]/\gamma$ can be expressed by a product of finite abelian ...
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49 views

Solvable group of order $p^nq^m$

Let $G$ be a semi-direct product of a $p$-group and a $q$-group where $p$ and $q$ are prime number. If $G$ does not contain a normal minimal subgroup of order $q$ what we can say about $q$-sylow of ...
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43 views

inner automorphisms of non-abelian simple groups

Let $G$ is non-abelian and simple group. Let $I ={\rm Inn}(G) \cong G$, $A = {\rm Aut}(G)$ and $B = {\rm Aut}(A)$. Since $Z(A)=1$, we have $A \cong {\rm Inn}(A)$, so we can identify $A$ with the ...
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78 views

Is this problem still open or solved?

Problem- Let $n\geq$ and let $T$ be the set of all permutations in $S_n$ of the form $t_k=\prod_{1\leq i\leq k/2}(i,k-i)$ for $k=2,3,4.....(n+1)$. Then find the least integer $f_n$ such that ...
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30 views

Inverse of zero missing for all finite fields F2

I am having a little touble with finite fields at the moment. I am just working from a high school text wich says that the inverse of an element in a group is unique, which to me implies that all ...
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57 views

Supersolvable groups of order $pq^m$

According to Burnside's classification in his book "Theory of groups of finite order", one of the types of non-abelian groups of order $pq^2$ ($p$ and $q$ are distinct primes), has the presentation ...
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55 views

Group theory question (on Nilpotent Groups)

use this notation for the following $\textbf{Theorem}$ - $\textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$ $\textbf{Theorem}$- In a finitely generated ...
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33 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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48 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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24 views

How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
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36 views

Two Open Ended Questions in Sylow Theory

Sylow Theorems are very powerful in finite group theory. Two natural questions come to mind: 1) Given a finite group $G$ and a $p$-subgroup $H$ of $G$, how many Sylow $p$-subgroups of $G$ contain ...
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64 views

Automorphism Tower

Let $G$ be a group. Now consider $\operatorname{Aut} G$ then $\operatorname{Aut} (\operatorname{Aut} G) = \operatorname{Aut}(2G)$, then $\operatorname{Aut}(\operatorname{Aut}(\operatorname{Aut} ...
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31 views

Finding commutator of $\mathbb D_n$ and $\mathbb S_4$

I have to find $[G,G]$ (the commutator of $G$) for the the dihedral group, $\mathbb D_n$ ; and the symmetric group, $\mathbb S_4$. What tools, theorems could I use in order to find these two ...
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42 views

Relation between $|H \lor K|$ , $|H|$ and $|K|$

Let $H$ and $K$ be subgroups of a finite group , then we know that the subgroup generated by $H \cup K$ i.e. $H \lor K$ is the smallest subgroup containing both $H$ and $K$ , then how can we relate ...