Tagged Questions

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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-4
votes
1answer
279 views

Topics in Algebra by I.N.herstein

If $G$ is finite cyclic group, show that $G$ cap is cyclic and $O(G\ \mathrm{cap})=O(G)$, hence $G$ and $G$ cap are isomorphic
4
votes
2answers
197 views

Show that the group $G$ is of order $12$

I am studying some exercises about semi-direct product and facing this solved one: Show that the order of group $G=\langle a,b| a^6=1,a^3=b^2,aba=b\rangle$ is $12$. Our aim is to show that ...
3
votes
1answer
156 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
3
votes
4answers
429 views

The product of all elements in $G$ cannot belong to $H$

Let $G$ be a finite group and $H\leq G$ be a subgroup of order odd such that $[G:H]=2$. Therefore the product of all elements in $G$ cannot belong to $H$. I assume $|H|=m$ so $|G|=2m$. Since ...
7
votes
2answers
638 views

Every normal subgroup of a finite group is contained in some composition series

In this context composition series means the same thing as defined here. As the title says given a finite group $G$ and $H \unlhd G$ I would like to show there is a composition series containing $H.$ ...
7
votes
0answers
80 views

Fixed points of coset operation

Let $G$ be a finite group which operates on two finite sets $E_1$ and $E_2$. Say that $E_1$ and $E_2$ are weakly $G$-isomorphic if for every $g \in G$, $\mathrm{Card}(E_1^g)=\mathrm{Card}(E_2^g)$, ...
6
votes
1answer
161 views

$ K(G)=3 \Longrightarrow G\cong\mathbb Z_3\ \mathrm{or} \ G\cong S_3$

According to J.S. Rose book "A Course on Group Theory": In class equation $$|G|=\sum_{i=1}^k|G:C_G(x_i)|$$ where $x_1,x_2,...,x_k\in G$ one from each of above $k$ classes; $K(G)$ is called the ...
3
votes
0answers
161 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
1
vote
1answer
186 views

$p$-Sylow in quotient groups

Prove that if $P$ is a $p$-Sylow of $G$ and $N \triangleleft\> G$ then: $ PN/N $ is q $p$-Sylow of $G/N$ $P \cap N $ is $p$-Sylow of $N$
1
vote
1answer
320 views

group order $p^2q^2$ will be abelian

let $(G,*)$ a group order $p^2q^2$ such that $q\nmid p^2 -1 $ y $p\nmid q^2 -1$ then $G$ is abelian. for Sylow theorem $n_p\equiv 1\mod (p)$ then $n_p = 1, q, q^2 $ but $n_p\neq p^2$ the same form ...
10
votes
1answer
120 views

Do all representations of finite groups have one-dimensional subrepresentations?

Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $$u = \sum_{g\in G} gv.$$ Then for any $g\in G$ we have $gu = u$ and therefore $<u>$ is a subrepresentation of ...
5
votes
2answers
169 views

$p$ is prime and $p^2\large\mid\normalsize|G|$

Hints needed: Let $p$ be a prime and $G$ a finite group such that $p^2\large\mid\normalsize|G|$ then $p\large\mid\normalsize|\text{Aut}(G)|$.
2
votes
2answers
290 views

A non-abelian $p$-group $G$

There is some facts about finite non abelian $p$-groups over the site. For example, when $n=3$: Nonabelian groups of order $p^3$. I have found the following problem in my very old works unsolved, ...
5
votes
2answers
766 views

When does an abelian group have a composition series?

There is an exercise in the book "An Introduction to the group theory by J.J. Rose" which can also be found as a proposition in "Abstract algebra by T. Hungerford": Every finite group has a ...
8
votes
2answers
235 views

Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the "irreducible" part isn't important, ...
6
votes
1answer
491 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
0
votes
1answer
226 views

Semidirect products of an elementary abelian p-groups and cyclic groups of prime order

(1) If $A$ is a elementary abelian p-group. And $Q=\langle t\rangle$ is a group of order q ($q\neq p$ prime numbers). For which primes p,q does the semidirect product $A\rtimes Q$ exist (so ...
2
votes
0answers
127 views

Automorphisms of a group and cyclic subgroups

I have the following question: Let $G$ be a group. And $A,B\leq G$ two cyclic subgroups with $|A|=|B|=n$. When does an Automorphism $\alpha\in\mathrm{Aut}(G)$ exists such that $\alpha(A)=B$? If ...
2
votes
1answer
189 views

conjugacy classes in representation theory

I have a question on conjugacy classes in this post, especially to this sentence: "if $g$ is a rotation of order $5$, then it is $K$-conjugate to each of $g^{13},g^{13^2},g^{13^3},g^{13^4}=g$". ...
1
vote
1answer
291 views

program to find matrix group given generators (Matlab)?

Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group? The best I've ...
1
vote
1answer
155 views

$H\trianglelefteq G$ is maximal then $H=C_G(H)$

Let $p$ is a prime and $G$ is a finite $p$-group. Also, the normal subgroup $H$ of $G$ is maximal among abelian subgroups of $G$ which are normal in $G$ as well. Prove $H=C_G(H)$. I should ...
3
votes
3answers
84 views

How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
6
votes
4answers
220 views

$p\mid [G:H]$ then $p\mid [N_G(H):H]$

I encountered the following problem for the first time. I sketched a proof for it. I will be thankful if I know it is correct or not. Thanks. $p$ is a prime and $H$ is a $p$-subgroup of a finite ...
2
votes
2answers
62 views

$H\vartriangleleft G$ and $|H|\not\equiv 1 (\mathrm{mod} \ p)$ then $H\cap C_{G}(P)\neq1$

Let $G$, a finite group, has $H$ as a proper normal subgroup and let $P$ be an arbitrary $p$-subgroup of $G$ ($p$ is a prime). Then $$|H|\not\equiv 1 (\mathrm{mod} \ p)\Longrightarrow H\cap ...
2
votes
0answers
78 views

Connection between: Expand a representation and dual representation

I worked the proof for 3) in this link out, but I have problems with the last step: Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual ...
2
votes
1answer
213 views

Number of elements of order $7$ in a group of order $28$

Given a group $G$ with order $28 = 2^2 \cdot 7$. Sylow-Theory implies that there is a exactly one $7$-Sylow-Subgroup of order $7$ in $G$, and $1$ or $7$; $2$-Sylow-Subgroups. Where to go from here ...
11
votes
2answers
302 views

$G/H$ is a finite group so $G\cong\mathbb Z$

Let $G$ is an abelian infinte group such that for all nontrivial subgroups $H$ $$\forall H\leq G, \left|\frac{G}{H}\right|<\infty$$ Prove that $G\cong\mathbb Z$. What I have done: Clearly, it ...
8
votes
2answers
336 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
2
votes
1answer
172 views

Question of Clifford theory

I have some questions about thist post: faithful irreducible representations of cyclic and dihedral groups over finite fields I would appreciate it really if someone could help me. 1) Do I get with ...
2
votes
4answers
236 views

$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.

I've tried, but I can't solve the question. Please help me prove that: $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
3
votes
1answer
222 views

Why are all groups of order 153 abelian?

$153 = 3^2 \cdot 17$ so lets assume there are $s_3$ $3$-Sylow-Subgroups and $s_{17}$ $17$-Sylow-Subgroups. We know that $s_3 \mid 153$ so $s_3 \in \{1,3,9,17,51,153\}$ and $s_{17} \in \{1,17,51\}$. ...
5
votes
2answers
640 views

Non-abelian group $G$ of order $p^3$

I just need some hints to prove this: Let $|G|=p^3$ be a a non-abelian group. If every subgroup of $G$ is normal, then $p=2$ and $G=Q_8$. I know the following facts about a non-abelian group ...
5
votes
1answer
115 views

Is this problem correct that $HG'=G$?

Here, I have the following homework: Let $G$ is a finite $p-$group and let $H$ be a subgroup of it such that $HG'=G$. Prove that $H=G$ ($G'$ is the commutator subgroup). I have tried to show ...
2
votes
1answer
244 views

Converting GAP groups into SAGE permutation groups.

I have been working with SAGE online, and have made some programs to test some hypothesis about finite groups. However, the pre-defined "named" groups in SAGE are quite limited (basically, the ...
1
vote
4answers
84 views

$(p,\frac{n}{p^\alpha})=1$ then $p\nmid\binom{n}{p^\alpha}$

Let $n=p^\alpha m$ wherein $(p,m)=1$. Then we have $$p\nmid\binom{n}{p^\alpha}$$ What I have done is just playing with $\binom{n}{p^\alpha}$ ...
3
votes
2answers
286 views

An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
7
votes
1answer
446 views

Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.

I have been trying for a while to make sense of Ex V.3.5 & Ex III.10.1 in Brown's book 'Co-homology of Groups': Calculate the Co-homology of $S_3$ with co-efficients in $\mathbb{Z}$, possibly ...
6
votes
0answers
297 views

Resource: Group Theory

There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory. I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
3
votes
2answers
64 views

$P\in \operatorname{Syl}_p(S_n)$ implies that $P\in \operatorname{Syl}_p(A_n)$ and $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$

$\newcommand{\Syl}{\operatorname{Syl}}$ This is an exercise (with hint about the second part) in my own language book in Group theory, however, maybe it is a lemma or theorem in an standard book ...
2
votes
1answer
59 views

The $p^k$-rank of a subgroup is no greater than the $p^k$-rank of the group.

Recently I was given a handout containing (roughly) the following text: Let $A$ be a finite abelian group, and $p^k$ a prime power. The $p^k$-rank of $A$ is defined to be ...
0
votes
1answer
56 views

What is subgroup $\langle g^d \rangle$?

According to the introductory abstract algebra, it says that $g^n =e$ where $g$ is an element of some finite group. Then, it talks about the subgroup $\langle g^d \rangle$. What is it exactly, and ...
4
votes
1answer
247 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
3
votes
1answer
274 views

exponent of an abelian group

Let $p$ be a prime. Let $H_{i}, i=1,...,n$ be normal subgroups of a finite group $G$. I want to prove the following: If $G/H_{i}$, $i=1,...,n$ are abelian groups of exponent dividing $p−1$, then $G/N$ ...
4
votes
1answer
233 views

Abelian subgroup of a group of order $2002$

Another unsolved question from my studying for quals - Show that if $G$ is a group of order $2002=2\cdot 7 \cdot 11 \cdot 13$, then $G$ has an abelian subgroup of index 2. I know it has to do with ...
5
votes
1answer
205 views

Complexity of finite group isomorphism problem

Consider the next decision problem: Given two finite groups represented by their multiplicity table, determine if they are isomorphic or not. Clearly, this problem belongs to NP since given a witness ...
4
votes
0answers
93 views

In finite groups does counting orders of elements is enough to determine if they are isomorphic [duplicate]

Possible Duplicate: Three finite groups with the same numbers of elements of each order Suppose that we have two finite groups $G$ and $H$ such that for each $n\in\mathbb{N}$ ...
8
votes
2answers
312 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
2
votes
1answer
239 views

Contragradient representation of a finite group

I am reading Serre's Linear Representations of Finite Groups and in an exercise in there he asks to show if $\rho$ is a representation of a finite group on $\textrm{GL}(V)$ with $V$ a finite ...
4
votes
1answer
554 views

Group cohomology of finite groups

I wonder if the group cohomology of a finite group $G$ with coefficients in $\mathbb{Z}$ is finite. This statement may be too strong. I am interested in, for instance, dihedral group. $$ ...
6
votes
1answer
180 views

Uniqueness of conjugates of a subgroup.

This question is partly influenced by the question: Are two subgroups that contain a common element conjugate iff they are conjugate under the normalizer? If we have an arbitrary finite group $G$ ...