Questions tagged [finite-groups]

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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Property of transfer homomorphism

Let $\pi$ be a set of primes, let $P$ be a Hall $\pi$-subgroup of a finite $G$ and let $G'(\pi)$ be the inverse image of $O_{\pi'}(G/G')$ in $G$. Consider the subgroup $P^*=\langle[y,g]:y,y^g\in P, g\...
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Modular group with a finite order $T$

Let $G$ be the modular group. We know this can be described by the relations (in terms of the $S$ and $T$ transformations) given by $S^4 = I, (ST)^3 = S^2$. In my work matrix representations of $G$ ...
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2 votes
2 answers
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Let $G$ be a group of order $30$ and $A, B$ be two normal subgroups of $G$ of order $2$ and $5$ respectively. Show that $G/AB$ contains $3$ elements.

The actual question is Let $G$ be a group of order $30$ and $A, B$ be two normal subgroups of $G$ of order $2$ and $5$ respectively. Show that $G/AB$ contains $3$ elements. In my approach: Firstly I ...
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$G'(\pi)$ is the smallest normal subgroup with an Abelian $\pi$-factor group.

I'm trying to show that following claim stated in Kurzweil and Stellmacher: Let $\pi$ be a set of primes, let $P$ be a Hall $\pi$-subgroup of $G$ and let $G'(\pi)$ be the inverse image of $O_{\pi'}(G/...
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$O_{\pi}(G/N)=PN/N$

Is the following statement correct? Let $\pi$ be a set of primes and let $P$ be a Hall $\pi$-subgroup of $G$. If $N$ is a normal subgroup of $G$ such that $G'\subseteq N$, then $O_{\pi}(G/N)=PN/N$. ...
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8 votes
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165 views

Are groups always isomorphic to their image under power maps?

Let $(G, \cdot)$ be a finite group of order $n$. Consider the map: $$G \to G, g \mapsto g^k$$ for $k$,$n$ coprime. This is injective, but generally not a homomorphism. Define a new group $G_k = (G,\...
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Group of the form $\langle a,b\mid a^4 =b^4=1 = (ab)^2,ab^3=ba^3\rangle$

I have seen the notion of a group $$G_{4,4}=\langle a,b\mid a^4 = b^4=1 = (ab)^2, ab^3=ba^3\rangle$$ of order $16$. Is there any generalization of this structure to any arbitrary $n$ i. e. $$G_{n,n}=\...
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Why need of totient function to find generators of cyclic group $C_n$?

Cyclic groups are defined where repeated application of the group operation is applied, to generate whole group, to generator element. But, have confusion as have seen questions that ask about ...
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If $G$ is nonabelian & solvable s.t. the centralizer of each nontrivial element is abelian, then $G$ is Frobenius with kernel its Fitting subgroup

I'm dealing with the following problem in Isaacs Finite Group Theory [6A.5], I would appreciate if you could help: Let $G$ be a nonabelian solvable group in which the centralizer of every nonidentity ...
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1 answer
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If $H\trianglelefteq K$ and $K\text{ char } G$, then is $H\trianglelefteq G$? [duplicate]

Let $G$ be a finite group and $H,K$ be subgroups of $G$ such that $H$ is normal in $K$ and $K$ is characteristic in $G$. Is $H$ normal in $G$? I know that if $H$ is characteristic in $K$ and $K$ is ...
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Subgroup generated by subgroups $\langle A_1,B_1\rangle$ and $\langle A_2,B_2\rangle$

Let $A_1$, $A_2$, $B_1$, $B_2$ be subgroups of finite group $G$. It is true that $$\langle\langle A_1,B_1\rangle,\langle A_2,B_2\rangle\rangle=\langle\langle A_1,A_2\rangle,\langle B_1,B_2\rangle\...
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1 answer
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Characterization of normal $p$-complements

Let $p$ be a prime number. $N$ is a normal $p$-complement of a finite group $G$ if and only if $O_{p'}(G)=N=O^p(G)$. Here $O_{p'}(G)$ is the largest normal subgroup whose prime divisors lie in $\...
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Let $H$ be subgroup of finite group $G$. Let $O(G)$ and $O(H)$ denote their order. If $O(G)/O(H)$ is prime, then is $H$ normal in $G$? [closed]

Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $O(G)$ and $O(H)$ denote the orders of $G$ and $H$ respectively. Identify which of the following statements are necessarily true. If $O (G) ...
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2 votes
1 answer
48 views

Proof of Baer's theorem

Baer's Theorem: Let $x$ be a $p$-element of a finite group $G$. Suppose that $\langle x,x^g\rangle$ is a $p$-subgroup for every $g\in G$. Then $x\in O_p(G)$. Here, $O_p(G)$ denotes the largest normal ...
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Structure of general linear group and special linear group [closed]

Why general linear group and special linear group over a finite field are never simple
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Does order of a cyclic subgroup of a finite group divide the order of this finite group?

I'm now reading J.Rotman's "A First Course in Abstract Algebra, 3rd edition". I am confused about Proposition2.73 in page 149. Proposition 2.73. A group $G$ of order $n$ is cyclic if and ...
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Dilation Properties of the Discrete Fourier Transform

I am working through Audrey Terras's "Fourier Analysis on Finite Groups and Applications" and am confused about one of the exercises in chapter 2. Towards the bottom of page 45, there is an ...
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3 votes
1 answer
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Proof explanation: 1.7.3 from Kurzweil and Stellmacher

I have a doubt in the following proof from Kurzweil and Stellmacher (transcribed below): Why is every normal subgroup of $E_1$ also normal in $N$? My attempt: For simplicity, assume $n=2$ and let $H\...
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For finite groups, $[F(G),E(G)]=1$

Let $G$ be a finite group, $F(G)$ be the Fitting subgroup of $G$ and $E(G)$ be the layer of G (subgroup generated by components of $G$). Then $[F(G),E(G)]=1$. My attempt: Since $E(G)$ is in ...
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If $EZ/Z$ is a component of $G/Z$, then $E'$ is a component of $G$.

I have a doubt in the proof of the following given in Kurzweil and Stellmacher: Let $Z$ and $E$ be subgroups of a finite group $G$ such that $Z\leq Z(G)$ and $EZ/Z$ is a component of $G/Z$. Then $E'$ ...
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1 vote
0 answers
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Fusion rules for irreps evaluated at different group elements?

It is well-known that tensor products of irreducible representations of a finite group decompose into direct sums of irreducible representations according to fusion rules $$\Gamma_i \otimes \Gamma_j=\...
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29 views

Can we extend this definition of graph of group?

Let $G$ be finite nonabelian group. According to Aalipour et al. (2016), they defined the enhanced power graph of $G$ as a simple undirected graph where the vertices are all elements of $G$ and two ...
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0 votes
2 answers
56 views

If $G=AB$ and $|G|=|A||B|$, then $G=A^{-1}B$.

Let $A$ and $B$ be subsets of a finite Abelian group $G$ such that $G=AB$ and $|G|=|A||B|$. Show that $G=A^{-1}B$. My attempt: If $A$ is a subgroup, then $A=A^{-1}$ and we're done. If $A$ is not a ...
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3 votes
2 answers
116 views

If $G=\langle H,K\rangle$, then $G'=[H,K]$

Let $H,K$ be two Abelian subgroups of a finite group $G$ such that $G=\langle H,K\rangle$. Show that $G'=[H,K]$. My attempt: Ofcourse, $[H,K]\subseteq [G,G]=G'$. Conversely, let $x,y\in G$. It ...
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1 vote
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Given a finite group $G$ and $N$ a normal subgroup such that $N\cong K$ where $K$ is Klein group and $G/N\cong \Bbb{Z}/7\Bbb{Z}$, prove $N\leq Z(G)$

Given a finite group $G$ and $N$ a normal subgroup such that $N\cong K$ where $K$ is Klein group and $G/N\cong \mathbb{Z}/7\mathbb{Z}$, prove $N\leq Z(G)$ I managed to deduce from both of the ...
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1 vote
1 answer
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Group of isometries acting on a metric space is already discrete if a stabilizer is finite and an orbit is discrete

My question is on page 163, the proof of Lemma 7 in the book Foundations of hyperbolic manifolds by John G. Ratcliffe. Let $\Gamma$ be a group of isometries of a metric space $X$. If there is a point $...
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2 votes
1 answer
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Prove $G \cong H \times K$ when $H,K \lhd G$ and the orders of $H,K$ relatively prime with the product of their orders equaling the order of $G$

Problem Statement Suppose $G$ is a finite group. $H,K \lhd G$ normal subgroups, $\gcd(\lvert H \rvert, \lvert K \rvert)=1$ and $\lvert G \rvert = \lvert H\rvert \lvert K \rvert.$ Prove $G \cong H \...
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2 votes
1 answer
47 views

Existence of a finite simple group that satisfies particular properties.

Let $G$ be a finite simple group and $\tau_G = \{ o(x) : x \in G\}$. Does there exist $d_1, d_2 \in \tau_G$ that satisfy the following: $d_1 < d_2$ and $d_1$ does not divide $d_2;$ for $x, y \in ...
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4 votes
1 answer
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Problem 5.17, Isaac's Character Theory Of Finite Groups

I couldn't find how I should go to the result in the following problem. ( Problem 5.17, Isaac's Character Theory Book ) Let $H \leq G$ and let $\chi = (1_H)^{G}$. Fix a positive integer $n$. For $g \...
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0 votes
1 answer
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Proving cyclic subgroup of a finite group is finite

Prove that if $G$ is finite then $\forall a\in G$, $H:=\langle a\rangle$ is finite. Pf: Let $G$ be a finite group with order $m$ and let $a \in G$. Suppose that $\langle a \rangle$ is infinite then ...
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0 answers
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magma gives an order 0 group? [closed]

I want to create an elementary abelian subgroup C of order 16. But the following Magma code gives "#C = $0$"... Where did it go wrong? F:=GF(3); A:=AlgebraicClosure(F); a:=RootOfUnity(8,A); ...
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2 votes
0 answers
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Under which group conditions is the homomorphism $g_2 f_1 :H_1 \to G\to H_2$ an isomorphism?

Let $G$ be a group and $H_1 ,H_2$ be two r-images of $G$; i.e. there exist two homomorphisms $f_i :H_i \to G$ and $g_i :G\to H_i$ such that $g_i f_i =id_{H_i}$, for $i=1,2$. My question: Under which ...
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0 votes
1 answer
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Let $G_1,G_2$ be groups, $N$ a normal subgroup of $G_1\times G_2$. Suppose $p_i(N)=G_i$ for all $i\in\{1,2\}$. Does it follow that $N=G_1\times G_2$?

Let $G_1,G_2$ be groups and $N$ a normal subgroup of $G_1\times G_2$. Let $p_i\colon G_1\times G_2\to G_i$ be the projection with $p_i(g_1,g_2)=g_i$ for $i\in\{1,2\}$. Suppose $p_i(N)=G_i$ for all $i\...
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-2 votes
0 answers
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Prove $\operatorname{Dih}(n) \cong \langle a,b \mid a^2,b^n,abab\rangle$. [duplicate]

Prove $\operatorname{Dih}(n) \cong \langle a,b \mid a^2, b^n, abab \rangle$. My attempt: First, denote $X={\{a,b\}}, \; R = \langle\!\langle a^2, b^n, abab \rangle\!\rangle, \; G = F_X/R$. Let $\phi:...
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0 votes
0 answers
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Representation theory of Lie rings

A Lie ring $(M,+)$ is an abelian group with a product $[\ ,\ ]$ (termed as the Lie bracket) satisfying $[x,x]=0$ $[\ ,\ ]$ is bilinear $[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,y,z\in M.$ I want ...
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3 votes
1 answer
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If $G$ is a simple group, with $\chi \in{\rm Irr}(G)$, such that $\chi(1) = p$, for some prime $p$, then $G$ has a Sylow $p$-subgroup of order $p$.

I am currently reading "Character Theory of Finite Groups". If the Sylow $p$-subgroup $P$ is abelian, then by Theorem $3.13, p$ is the exact power of $p$ diving $|G: Z(G)| = |G|$ since $G$ ...
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3 votes
1 answer
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Given $p$ odd prime and $G=S_p$ and $P\leq G$ $p$-Sylow and $H=N_G(P)$ find $|H|$ and prove if $p=5$ then $H \cong C_4 \ltimes C_5 $

Given $p$ odd prime and $G=S_p$ and $P\leq G$ $p$-Sylow and $H=N_G(P)$ find $|H|$ and prove if $p=5$ then $H \cong C_4 \ltimes C_5 $. For the first part we know that if we set $n_p$ the number of ...
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  • 1,858
2 votes
0 answers
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Restrictions on the order of Finite Simple Groups

I've just started learning about simple groups and I'm curious about restrictions on the order of finite simple groups. For example, I know that the only abelian finite simple groups are cyclic with ...
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6 votes
0 answers
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Can the converse of Lagrange's Theorem hold for composite integers? [duplicate]

In general the converse to Lagrange's Theorem is false - $A_4$ has no subgroup of order $6$. However, the converse holds for the set of primes - given a prime $p$ and a finite group $G$, if $p$ ...
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1 vote
1 answer
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Finding maximal, solvable, primitive subgroups of a large group in GAP

I am trying to find the maximal, solvable, primitive subgroups of a large group $N$ which is itself a subgroup of $GL(n,p)$ for $(n,p)=(4,3),(4,5),(4,7),(6,3),(10,3)$. However, GAP is too slow to run ...
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2 votes
1 answer
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Prove a group of order $22,000$ with $16$ Sylow-$5$ subgroup has a normal (Sylow) $11$ subgroup.

Let $G$ be a group of order $2^4\cdot 5^3 \cdot 11$, $H$ be a group of order $5^3 \cdot 11$. Prove $H$ has a normal $11$-subgroup. Suppose $n_5(G) < 16$ (number of Sylow $5$-subgroups of $G$), ...
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0 votes
1 answer
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Let $G$ be a finite group, $|G|=n$, there are $x\in \mathbb{N}$ conjugacy classes. Prove there are $xn$ homomorphisms from $\mathbb{Z}^2$ to $G$.

Let $G$ be a finite group, $|G|=n$, there are $x\in \mathbb{N}$ conjugacy classes. Prove there are $xn$ homomorphisms from $\mathbb{Z}^2$ to $G$. My approach is to find out the numbers of the normal ...
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-4 votes
0 answers
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Prove that $\text{Hom}(G,C^*)$ is isomorphic to $G$ if and only if $G$ is abelian.

Let $G$ be a finitely generated group. Prove that $\text{Hom}(G,C^*)$ is isomorphic to $G$ if and only if $G$ is abelian. My try: Suppose that $\text{Hom}(G,C^*)$ is isomorphic to $G$. To show $xy=yx$ ...
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2 votes
1 answer
35 views

GAP Program Efficiency for Number of Orbits

Let $S$ be a nonabelian finite simple group and $p$ a prime divisor of $|S|$. I'm interested in finding the number of $\textrm{Aut}(S)$-orbits acting on the set $\textrm{Cl}_{p'}(S)$, the set of all $...
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0 votes
1 answer
63 views

Orbits of a matrix group

How to get the orbits of the action of the following matrix group on the standard basis of a 3-dim vector space? \begin{pmatrix} SL_2(2) & 0\\ * & 1\\ \end{pmatrix} where * denotes a 1$\times$...
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-2 votes
1 answer
86 views

Let $p$ be the smallest prime divisor of $|G|$ and $N\unlhd G $ s.t. $|N|=p$. Find ${\rm Aut}(N)$. [closed]

Let $p$ be the smallest prime divisor of $|G|$ and $N\unlhd G $ s.t. $|N|=p$. Find ${\rm Aut}(N)$. Attempt: We have $|N|=p$ and since $p$ is a prime number, $N$ is a cyclic subgroup and so abelian, ...
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4 votes
1 answer
90 views

On Alperin's paper "The Green Correspondence and Brauer's Characterization of Characters" (aka what is a central factor?)

I was studying the paper "The Green Correspondence and Brauer's Characterization of Characters" by J. Alperin and I couldn't understand two of the passages. Hypotheses and notations $G$ is a ...
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0 votes
1 answer
42 views

Why $D_4$ is the biggest group generated by relations $\langle f,r | f^2 =1 ,r^4 =1 , fr=r^3f \rangle$? [duplicate]

I want to find the presentation of group $D_4= \{1, f, r,r^2,r^3, rf, r^2f, r^3f \} $. $r$ is the rotation of a square counterclockwise by 90 degree and $f$ is the action that flips the square. Here $...
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1 vote
0 answers
33 views

Is any $S_4$-invariant function also $S_6$ invariant?

Consider the following embedding of the permutation group $S_4$ inside $S_6$: $\sigma \in S_4 \to \tilde \sigma \in S_6$, where $$ \tilde \sigma\big(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}\big)=\big(...
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0 votes
1 answer
38 views

Isomorphisms between groups of order 42 [duplicate]

I'm trying to prove that there are exactly 5 groups of order 42. My approach was to show that there is always a subgroup of order 6, let's say $H$, and a normal group of order 7, let's say $K$. It ...
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