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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3
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1answer
92 views

Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
2
votes
2answers
33 views

Finite groups and one-to-one functions on them.

I am having trouble with this problem: Assume that $(\mathbb{G}, *)$ is a finite group and there exists a positive integer $n$ such that gcd($n, |\mathbb{G}|)=1$. Prove that the function $F_n: \...
1
vote
1answer
113 views

Finite group $G$ is product of a subgroup $H$ and normalizer of a Sylow $p$-subgroup of $H$

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of $H$. Let $N_G(P)$ be the normalizer of $P$ in $G$. Show that $G=N_G(P)H$.
0
votes
1answer
96 views

Sylow subgroups of soluble groups

Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
1
vote
1answer
155 views

Elements of order 3 in $PGL(4,\mathbb{R})$

I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead $GL(4,\mathbb{C})$ ...
3
votes
0answers
149 views

Central extensions of elementary abelian p-groups

Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is : How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n \...
1
vote
2answers
62 views

primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
0
votes
0answers
68 views

Show that the number of elements in the equivalence class is n?

Question is : Let $G$ be a finiote group of order n = $p^{\alpha}$m, where $p$ is a prime number and if $p^r$ | m but $p^{r+1} \nmid$ m Let $\mathcal M$ be the set of all subset of $G$ which have $...
0
votes
0answers
52 views

About some special kinds of group automorphisms

let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$. Is there any non-trivial ...
5
votes
2answers
118 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
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vote
0answers
58 views

a question about fixed-point-free automorphism group 2

In this paper, Rowley (1995), there is a theorem: Let $A$ and $G$ be finite groups. Suppose that $A$ acts fixed-point-freely on $G$ and that either $A$ is cyclic or $(|G|,|A|)=1$. Then $G$ is ...
0
votes
1answer
69 views

Find the conjugacy classes of $D_6$

I am following an example in my lecture notes, but I have come to a part which I cannot get to work for myself. Thanks. Find the conjugacy classes of $D_6$. Take $$G = D_6 = \langle a,b \mid a^3 =b^2 ...
3
votes
1answer
860 views

Irreducible representations (over $\mathbb{C}$) of dihedral groups

Find number of complex irreps of the group $D_n$. Find dimension of the irreps. I know that The number of complex irreps of a finite group is equal to the number of conjugacy classes of the ...
1
vote
1answer
42 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a $G-$...
0
votes
1answer
231 views

Lower Exponent P Central Series

The lower exponent $p$-central series for a $p$-group $G$ is defined by $G=P_1(G) > P_2(G) > \ldots > P_c(G) = 1$, where $$P_i(G)=[P_{i-1}(G), G] P_{i-1}(G)^p.$$ If $G_i=G/P_i(G)$ and $A:G_{...
1
vote
1answer
67 views

GF(113) arithmetic using tables?

I need to work with the Galois Field of (prime) characteristic 113. I am wondering if it is possible to implement multiplication and division using log/antilog tables (like I already do in different ...
1
vote
2answers
146 views

the nilpotency class of Frobenius kernel

As we know, if G has a fixed-point-free automorphism of order p, then G is nilpotent, can we know something about the nilpotency class of Frobenius kernel ?
3
votes
1answer
193 views

a question about fixed-point-free automorphism

Let G be a finite group with a fixed-point-free automorphism a of order 3. Prove that [x,y,y]=1 for all x,y in G.
5
votes
1answer
226 views

Finite abelian p-group and an element of maximal order

I'm studying for an exam and I'm having trouble understanding the proof given for the following statement: Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there ...
4
votes
2answers
87 views

Find the center of a specific group

The group $G$ is generated by the two elements $\sigma$ and $\tau$, of order $5$ and $4$ respectively. We assume that $\tau\sigma\tau^{-1}=\sigma^2$. I have shown the following: * $\tau\sigma^k\tau^{-...
1
vote
1answer
57 views

Composition Series of $A_4 \times S_5$

Please help me with the following question: Find the composition series of $A_4 \times S_5$ and prove that this series is indeed a composition series. Afterwards, find a group with the same ...
0
votes
1answer
121 views

If $p$ is a prime number and $ \ \ p^{\alpha}|o(G)$, then $G$ has a subgroup of order $p^{\alpha}$

I know the proof of this result, but i have doubt in the proof of "I.N Herstein". Let $G$ be a group of order n = $p^{\alpha}$m, where $p$ is a prime number and if $p^r$ | m but $p^{r+1} \nmid$ m ...
5
votes
1answer
67 views

The Four Transitive Subgroups of $A_7$.

I know that $A_7$ contains 3 transitive subgroups: $A_7, PSL(2,7), F_{21}$ (Alternating group of 7 elements, $PSL(2,7)$, Frobenius group of order 21). In studying the Galois group structure of ...
3
votes
1answer
170 views

How to find multiplicative orders of all elements in field $\Bbb F$ (say $\Bbb F_{13}$)?

I am working on some finite fields and I was referring to some online class material. Is there any way to find the multiplicative orders of all elements in a field $\Bbb F$?
5
votes
4answers
149 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
6
votes
1answer
432 views

Prove that $H$ is a abelian subgroup of odd order

Question is: Let $G$ be a group of order 2n. Suppose half of the element of G are of order 2 and the other half form a subgroup $H$ of order n . Prove that $H$ is of odd order and is an abelian ...
0
votes
1answer
99 views

Prove that every element of a group G can be represented as $g = x^{-1}(xT)$ for some x $\in$ G? [duplicate]

Let G be a finite group and T be an automorphism on G with the property that T(x) = x for $ \ \ $ x $\in$ G iff x = e. Prove that every element of G can be represented as $g = x^{-1}(xT)$. Suppose ...
7
votes
3answers
194 views

What is the probability of product of two elements is desired element?

Let $G$ be a group with $n$ element. Fix $x\in G$. If you choose randomly two elements from $G$, what is the probability of $x$ being product of these two elements? At first, I thought answer was $1/...
0
votes
1answer
147 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
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vote
0answers
94 views

Is there an easy proof for the classification of $6$-transitive finite groups?

For the background, see the post: Classification of triply transitive finite groups Thanks to the classification of finite simple groups (CFSG), we know that if $G$ is a finite $6$-transitive ...
0
votes
1answer
53 views

Free action of finite direct product

Let's consider free action of finite abelian group $G = G_1 \oplus G_2$ on a manifold $X$. Is it true that $X/G$ is diffeomorphic to $(X/G_1)/G_2$?
2
votes
1answer
67 views

Abstract Algebra: Cosets

Find all of the distinct left cosets of <4> in Z18 and all the cosets of <4> in the subgroup <2> of Z18. So The distinct left cosets of <4> in Z18 are 0 + <4> and 1 + <4>. Do I ...
0
votes
0answers
45 views

A dense subset of a finite group

Let $G$ be a finite group with Zariski topology. Suppose $G=A_1\cup A_2\cup\cdots\cup A_n$, where $A_i$, $1\leq i\leq n$, are pairwise disjoint subsets of $G$ and only $A_1$ is dense in $G$, that is, $...
5
votes
2answers
501 views

Order of automorphism group

I have this tiny question that I just can't figure out: Let $G$ be the dihedral group of order 8. Show that Aut($G$) is a $2$-group. I know that there is a general way to calculate the order of the ...
2
votes
1answer
142 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
4
votes
0answers
375 views

Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$. Is ...
0
votes
2answers
116 views

A group with order 12 with three elements of order 2 [closed]

Show that $A_4$ (which has order $12$) has exactly three elements of order $2$. Additional information: $A_4$ denotes the set of even permutations in $S_4$. $S_4$ is defined as all of the ...
4
votes
2answers
2k views

Subgroups of the Klein-4 Group

Can anyone explain to me the subgroups of the Klein-4 group? I'm trying to view it this way: I want some groups that are not empty and $ab^{-1} \in H$, where $H$ denotes the subgroups I am looking ...
0
votes
0answers
29 views

Composition Factors of $C_p\times C_p$

I have question that asks me to find the composition series of $C_p\times C_p$, now these are all isomomrphic to the series $\{1\}\lhd C_p \lhd C_p\times C_p$ but the questions wants all the series ...
0
votes
1answer
114 views

Order of Sylow $p$-subgroups

My class is studying on Sylow $p$-subgroups, and I had been stuck for several hours on determining the order of a Sylow $p$-subgroup of a group $G$ of finite order. I asked a previous question like ...
0
votes
1answer
253 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order $2$ or that $G$ has more than one element of order $...
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vote
2answers
110 views

On the graph of induction-restriction for group-subgroup representations

Let $G$ be a finite group, and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.). Consider the graph $\mathcal{G}(...
4
votes
0answers
81 views

Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group $G_\...
2
votes
3answers
358 views

Show that no $U$-group of order $16$ is isomorphic to $\mathbb{Z}_4\oplus \mathbb{Z}_4$

$$U(n)=\{x : 0<x<n, \gcd(x,n)=1\}.$$ We are asked to show that no $U$-group of order $16$ is isomorphic to $\mathbb{Z}_4\oplus \mathbb{Z}_4$. (External direct product) I started calculating ...
1
vote
2answers
90 views

Subnormal versus quasinormal subgroups

Let $G$ be a group and $H$ a subgroup. $H$ is subnormal if it exists a finite normal chain from $H$ to $G$. $H$ is quasinormal if $HS=SH$ for all subgroup $S$ of $G$. If $G$ is a finite group, ...
6
votes
1answer
248 views

Qualifying Exam Question On Elementary Group Theory

Question. Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$. Let $x$ be n element of order $p$ in $G$. Assume that there exists an element $h\in G$ such that $hxh^{-1}=x^{10}$. ...
3
votes
1answer
120 views

On semi-direct product of groups

If for two finite groups $G$ and $H$ we have $G/N \cong H$, where $N$ is a normal subgroup of $G$, can we say $G\cong NH$ as a semidirect product?
7
votes
1answer
139 views

Reference request: Introduction to Finite Group Cohomology

I don't know anything about group cohomology and I'd like to. What is the best text to learn this subject? I'd prefer as soft an introduction as possible - that is, lots of motivation, lots of ...
8
votes
1answer
482 views

Finite groups with all maximal subgroups of prime power index are solvable?

It is well known that in finite solvable groups, any maximal subgroup has prime power index. Question: If $G$ is a finite group in which any maximal subgroup has prime power index, is $G$ ...
0
votes
1answer
66 views

character of a representation of the group $S_n$

Let $\Phi$ be a representation of the group $S_n$ in the space with basis $(e_1, \ldots, e_n)$ such that $\Phi(\sigma)e_i=e_{\sigma(i)}.$ Find character of $\Phi$ This is looks like a regular ...