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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2
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2answers
85 views

prove that group of order 275 has non trivial center.

Let $G$ be finite group of order $275 = 5^2\cdot11$. prove that $Z(G)=\{g\in G:\forall h\in G\space\space gh=hg\}\not=\{e\}$. Using the Sylow theorems I manged to prove that $G$ has normal subgroup ...
1
vote
0answers
28 views

General Classification of finite simple ternary groups?

Define a ternary group as an algebraic set endowed with a 3-ary operation f: that maps 3 elements onto another in the set. Furthermore for any three elements a,b,c there exists a unique 4th element ...
4
votes
1answer
124 views

Finite groups with nontrivial outer automorphisms

Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial. Question: Does there always exist an $f \in \text{Aut}(G)$ ...
2
votes
1answer
55 views

Order of Aut$(D_4)$

How can I prove that order of Aut$(D_4)$ is 8. Let we show $D_4$ as $\{e,\sigma,\sigma^2,\sigma^3,\tau,\tau\sigma,\tau\sigma^2,\tau\sigma^3\}$ and ...
3
votes
1answer
36 views

Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
2
votes
1answer
46 views

$G=HK$ then the index of a subgroup is determined by $H$ and $K$

Let $G=HK$ s.t. $H\cap K=1$ and let $R$ be a any subgroup of $G$. I wonder necessary and suffucient condition for the equality, $$|G:R|=|H:H\cap R||K:K\cap R|$$ Note that if $H$ and $K$ are normal ...
2
votes
1answer
73 views

How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is ...
1
vote
0answers
32 views

Independent components of a group cocycle

Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all ...
2
votes
1answer
51 views

Number of cyclic subgroups order $p^2$ in $\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2}$

Let $$G={ {<a>}_{p} \times {<b>}_{p} \times {<c>}_{p^2}} \cong \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2} \text{, $p$ is prime}$$ There are $p^3-1$ elements with order ...
4
votes
1answer
90 views

Can this lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Let the lattice $\mathcal{L}$ as follows: ...
1
vote
1answer
33 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
1
vote
1answer
32 views

$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
1
vote
0answers
53 views

Show that the group of permutations of {1,2,3,4} is equal to the product of it's subgroups…

Show that the group of permutations of {1,2,3,4} $$\sigma_4$$ is equal to the product of it's subgroups $$C_2\times C_2 $$ and$$D_6=(x^3=y^2=1, yx=x^2y)$$ I'm not sure whether to just multiply the ...
0
votes
1answer
41 views

Finding unique groups of C11 semidirect product C5

Find unique groups of form $$ C_{11}\rtimes C_5$$ (semi-direct product) with homomorphism $$h:C_5\rightarrow Aut(C_{11})$$ I've found the possible homomorphisms i.e $$h=Id,x^3,x^4,x^5,x^9 $$ So ...
3
votes
1answer
57 views

Prove that the subgroups of G have order 2

Let $G = \{1_G, g, h, k\}$ be a group with $4$ elements and suppose $G$ is not cyclic. Using Lagrange’s Theorem show that $g$, $h$ and $k$ all have order $2$ and write down a table for the group ...
7
votes
1answer
90 views

How to humanly verify $ba^5ba=b^2a^4$ in the group with presentation $\langle a,b : a^7=1, b^3=1, ba^2=ab \rangle$?

The group with presentation $\langle a,b : a^7=1, b^3=1, ba^2=ab \rangle$ is isomorphic to $\mathbb{Z}_7 \rtimes \mathbb{Z}_3$ (ref.). Q: How can I (as a human) verify that $ba^5ba=b^2a^4$ in this ...
2
votes
2answers
91 views

Does a finite group always contains subgroups generated by its elements

Does a finite group contains subgroups generated by its elements. I suspect that answer is Yes and that my question could be trivial. But I am studying algebra for the first time so I wanted to be ...
1
vote
0answers
56 views

A group $G$ is said to satisfy the permutizer condition in $G$ if $P_{G}(H)$ strictly contains $H$ for any subgroup $H$ of $G$.

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with$H$, i.e. $P_{G}(H)= \langle x \in G | \langle x\rangle H = H \langle x ...
2
votes
1answer
40 views

Question about Permutation Sets (Groups and Symmetries)

Let $a = (123)(456)$ in $S_{10}$. Find the highest possible order of a permutation $b$ in $S_{10}$ such that $b^k=a$ for some $k$. Attempt: I already know that there are only two possible cases for ...
1
vote
1answer
88 views

If $ G$ is a finite group of order $n=|G|,$ then $a^n = e $ for all $a$ in $G $?

I.N. Herstein's book has a theorem (Theorem 2.4.5 Abstract Algebra, Second Edition) " If $ G$ is a finite group of order $n=|G|,$ then $a^n = e $ for all $a$ in $G $ ". However, for $\mathbb Z_6, ...
1
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1answer
52 views

Fit_{p}(G)$ equals the intersection of the centralizers of the principal factor of G whose order is divisible by p.

A finite group $G$ is said to be $p$-nilpotent (where $p$ is a prime) if it has a normal Hall $p'$-subgroup, that is, if $O_{p'p}(G) = G$. Obviously every finite nilpotent group is $p$-nilpotent; ...
4
votes
2answers
63 views

$N$ be a normal subgroup of $G$ such that $G/N=Aut(N)$

Let $N$ be a normal abelain subgroup of $G$ such that $N=C_G(N)$ and $G/N\cong Aut(N)$. Can we say that $G$ is a semidirect product of $N$ and $Aut(N)$ ?
1
vote
1answer
61 views

Requirement for Hall-$\pi$-subgroups to be conjugate (Wielandt)

I am reading through Isaacs Theory of finite groups and towards the end of chapter 3 on "split extensions" there is mention of a theorem of Wielandt which states that if a finite group $G$ contains a ...
2
votes
1answer
106 views

Commuting elements and conjugacy classes

Let $G$ be a finite group and $x\in G$ be an element of order $p$ ($p$ prime). Suppose that $x\in P$, where $P$ is some $p$-Sylow subgroup of $G$. I could not prove the following: $x$ is not ...
5
votes
0answers
194 views

Given an irreducible representation, is there a *unique* unitary representation that it is equivalent to?

I might need help here in understanding my own question in places and please don't hesitate in asking for edits and clarifications. Background: A representation $\rho$ of a finite group $G$ is a ...
1
vote
1answer
72 views

Interpreting a group homomorphism $f: \mathbb{Z}_{12} \to \mathbb{Z}_{3}$ visually

I am having a hard time studying and I am a visual learner. How could I visually imagine a (group) homomorphism $$\mathbb{Z}_{12} \to \mathbb{Z}_3?$$ Also, if the question states that the map $f$ is ...
1
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1answer
46 views

If [H,N]≠1 then H∩N≠1 and N≤H . Now NOp′(H) ,so that N∩Op′(H)=1. Hence N is a p-group. Now can we say CH(N)≥Op′(H) and H/CH(N)is p-group?

If $G$ is a group. Let $H$ be a normal $p$-nilpotent subgroup of $G$ and let $N$ be a minimal normal subgroup of $G$ whose order is divisible by $p$. If $[H,N]\neq 1$ then $H \cap N \neq 1$ and $ N ...
2
votes
1answer
102 views

Show that there is exactly one binary operation making the set $\{e, x, y\}$ a group with $e$ the identity element

Let $S=\{e, x, y\}$ be a set of three elements. Show that there is exactly one binary operation making the set $S$ a group such that $e$ is the identity element. So I have produced the ...
4
votes
1answer
96 views

When is a power map a homomorphism?

Let $G$ be a finite group and define the power map $p_m:G\to G$ for any $m\in\mathbb Z$ by $p_m(x)=x^m$. When is this map a group homomorphism? The $p_m$ is clearly a homomorphism whenever $G$ is ...
2
votes
3answers
98 views

Representation of Quaternion group in $GL(2,3)$

I am working with the representation of the quaternion group in $GL(2,3)$ generated by $A=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, B=\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}, ...
1
vote
2answers
50 views

If $G$ is a group. Let $H$ be a normal $p$-nilpotent subgroup of $G$ and let $N$ be a minimal normal subgroup of $G$

If $G$ is a group. Let $H$ be a normal $p$-nilpotent subgroup of $G$ and let $N$ be a minimal normal subgroup of $G$ whose order is divisible by $p$. If $[H,N]\neq 1$ then $H \cap N \neq 1$ ?? This is ...
2
votes
1answer
43 views

$G=H_1 \cup H_2 \cup H_3$ index of $H_i$

Let $G=H_1 \cup H_2 \cup H_3$ be a finite group, where each $H_i$ is proper subgroup of G. (I can show) $H_i \neq H_j$ where $i\neq j$ Show that each $H_i$ has index two in G Any suggestion?
3
votes
1answer
71 views

Proof of Hall's subgroup Theorem

So I'm working through Hall's Theorem for Solvable groups and there is one part of it which I cannot seem to prove. I am following through Isaac's book on Finite group Theory for reference. Currently ...
3
votes
0answers
104 views

$\gcd(|G|, |\text{Aut}(G)|)=1$ means G is abelian?

Prove the following assuming that $G$ is finite group with $\gcd(|G|, |\text{Aut}(G)|)=1$. a) G is abelian (done). b) Every Sylow subgroup of $G$ is cyclic of prime order. Since G is ...
1
vote
1answer
48 views

Is there any group in which number of the normal subgroups is equal to number of the conjugacy classes?

Let $G$ be a group s.t. $|G|\geq 3$. Is there any example of $G$ such that number of the normal subgroups is equal to number of the conjugacy classes?
2
votes
3answers
109 views

Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$

Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$ (the group with $x^3=1$, $y^2=1$ and $xy=yx^2$). I'm not really sure how to express the elements of $\operatorname{Aut}(C_2 \times ...
6
votes
0answers
86 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
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0answers
42 views

I need example to satisfy in this theorem (Hall Subgroup)

I need example to satisfy in this theorem: let $H$ be a subgroup of $G$ such that $\mid G : H \mid$ is a $\Pi$-number.If there is a nilpotent subgroup $K$ of $G$ such that $G=HK$ then $G=HK_{\Pi}$, ...
3
votes
1answer
57 views

Sylow subgroup of a group

Let $G$ be a group such that $\vert G\vert=231$. I have to show that the unique Sylow 11-subgroup of $G$ is contained in the center of $G$ I proceed as follows: Since the number of Sylow 11-subgroup ...
3
votes
0answers
64 views

Group of order 112

Let $G$ be a finite group of order $2^4\times 7$ and Sylow $7$-subgroup of $G$ is not normal. Prove that Sylow $2$-subgroup of $G$ is abelian. I am very grateful for any help in this problem.
7
votes
1answer
123 views

Do these groups have a meaning?

Let $G$ be a group. We can say that $Aut(G)\leq S_G$ where $S_G$ denotes the set of all bijection from $G$ to $G$. But $S_G$ is not a good bound for $Aut(G)$ as $S_G$ grows very fast. Let's define ...
1
vote
2answers
82 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
2
votes
1answer
42 views

Is this extension of $Sp(4,2)$ a semidirect product?

Somebody I trust has been insisting to me that a certain extension of $Sp(4,2)$ is actually a semidirect product, and I'm inclined to believe him, but I haven't been able to convince myself he's ...
2
votes
1answer
40 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
4
votes
2answers
131 views

Exercise 1: Galois Theory (J. Rotman)

Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...
2
votes
1answer
40 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
1
vote
1answer
33 views

Primitive Roots Modulo $2^n$ for $n\geq3$

Question: (a) Prove that there is no primitive root modulo $2^n$ for any $n\geq3$, where $\bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is a primitive root modulo $2^n$ if the order of $\bar{a}$ is ...
2
votes
1answer
82 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
2
votes
1answer
61 views

A normal intermediate subgroup in $B_3$ lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in $B_3$ lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...
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vote
0answers
105 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...