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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0answers
180 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
110 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
738 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
26 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
81 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
228 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 2. ...
1
vote
2answers
98 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 ...
2
votes
0answers
69 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 ...
2
votes
3answers
225 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
59 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
214 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
101 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?
5
votes
1answer
95 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 ...
7
votes
1answer
337 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
59 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 ...
9
votes
3answers
576 views

If H is a subgroup of G, then H has no more Sylow subgroups than G

If $H$ is a subgroup of the finite group $G$, then how do I show that $n_p(H) \leq n_p(G)$? Here $n_p(X)$ is the number of Sylow $p$-subgroups in the finite group $X$. Here is my attempt: Suppose ...
5
votes
1answer
309 views

groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
1
vote
1answer
51 views

Regarding the order of elements in a factor group

If I have understood it correctly, a factor group consists of all cosets of a subgroup $H$ of $G$. Since it is a requirement that $H$ is normal, left and right cosets are equivalent. I am asked to ...
1
vote
1answer
46 views

Identify a semidirect product $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$

I'm studying for the first time semidirect product and I'm trying to learn how to identify some of them. For example $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$ I red that, for ...
3
votes
1answer
147 views

A second isomorphism theorem for action on cosets II

Let $G$ be a finite group, and $K$, $L$ subgroups of $G$ and $H=K \cap L$ such that: $G = \langle K,L \rangle$. $\forall g \in G$ : $HgK=KgH$ and $HgL=LgH$ Remark: These assumptions imply that ...
2
votes
2answers
63 views

A second isomorphism theorem for action on cosets

Let $G$ be a finite group, and $K$, $L$ subgroups of $G$ such that $G = KL=LK$. Let $\Omega = G/K$ and $\pi: G \to S_{\Omega}$ the canonical action on cosets. Question: Is it true that $\forall ...
0
votes
1answer
46 views

Question on Nilpotentcy of $G$ given $G/N$ nilpotent

If $G$ is a group and $N<Z(G)$ with $G/N$ nilpotent then I want to show that $G$ is also nilpotent (here we take the definition of nilpotency for finite groups to be has a unique sylow subgroup of ...
0
votes
1answer
61 views

Number of homomorphisims from $C_5\times C_4\times C_4$ onto $C_{10}$

I know that any homomorphism between groups is determined by it's action a generating set of the group and that the kernel of such homomorphism must be of order 8 by the first isomorphism theory. By ...
2
votes
0answers
217 views

number of elements of each order in p-groups $Z_{p^n}\rtimes Z_p$ and $Z_{p^n}\times Z_p$ [closed]

Do $p$-groups $\mathbb{Z}_{p^{n}}\rtimes \mathbb{Z}_p$ and $\mathbb{Z}_{p^{n}}\times \mathbb{Z}_p$ have the same number of elements of each order? (The prime $p$ is odd.)
0
votes
1answer
43 views

Are all the groups of order $n$ contained in $S_n$?

I want to know if I can considere any group of order $n$ is isomorphic to one of $S_n$. Is that true? I can't find a proof.
0
votes
1answer
43 views

isomorphism between direct product of general linear groups

We know that for m=p_1p_2...p_n which p_i are prime numbers, then SL(2,Z_m) is isomorphic to direct productt of SL(2,p_i). Can we say that GL(2,2) is isomorph to direct product of itself ?
2
votes
1answer
47 views

Finding a particular chief series for a supersolvable group

Let $G$ be a finite supersolvable group. Since it is supersolvable it has (by definition) some normal series $1=N_0 \le N_1 \le \cdots \le N_n=G$ such that each factor $N_i/N_{i+1}$ is cyclic. Also ...
2
votes
2answers
75 views

How do I calculate permutations where some values are restricted?

I am curious about the formula for determining the number of combinations there are in a given set where some values are restricted to a certain range. For example, if I have a 10 character, ...
0
votes
1answer
43 views

direct product of general linear groups

How can we compute direct product of G with itself such that G=GL(2,2). We know that order of G is 6 and then the order of its direct product is 36. Since G is non-abelian, how can we describe the ...
0
votes
0answers
33 views

Is a finite group generated by a subset of order more than $n/2$? [duplicate]

Let $G$ be a group of order $n\in\mathbb{N}$ and $S\subset G$ a subset with $\# S>n/2$. How can I prove that $G$ is generated by $S$?
9
votes
1answer
124 views

if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
2
votes
0answers
18 views

Show that the order of the class of $p+1$ in $\left( \mathbb{Z}/p^{\alpha}\mathbb{Z}\right)^{*}$ is $p^{\alpha-1}$

I tried to do that: $$(1+x)^p=1+px+\binom{p}{2}x^2+\dots+\binom{p}{p-1}x^{p-1}+x^p$$ So if $x=0 \quad mod(p^k)$ then $(1+x)^p=1 \quad mod(p^{k+1})$ Now I'm trying to deduce that ...
3
votes
1answer
85 views

Order of a specific group

Is it true that the order of this group is $14$ (because of $7\cdot2$)? $$\langle S, T\mid S^7 = (S^4T)^4 = (ST)^3 = T^2 = 1\rangle$$
3
votes
1answer
33 views

Order of $\left(\mathbb{Z}/10\mathbb{Z}\right)^{*}$ and others groups

I already know that $\left(\mathbb{Z}/10\mathbb{Z}\right)^{*}=\left(\mathbb{Z}/2\mathbb{Z}\right)^{*}\times\left(\mathbb{Z}/5\mathbb{Z}\right)^{*}$ Theses groups have order 1 and 4 so the group is ...
2
votes
2answers
71 views

Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

I'm trying to determinate how many morphisms of groupes exist between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ for $n,m\in\mathbb{N}$. I know a morphism is determinated by the image of ...
3
votes
1answer
160 views

an order of automorphism group of finite abelian group

This is problem of Rotman's Exercise 7.9(i). If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order. How can I approach to this problem? Could you suggest ...
4
votes
1answer
58 views

Automorphisms of spin groups over finite fields, even dimension

I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of ...
1
vote
1answer
194 views

If $|A| > \frac{|G|}{2} $ then $AA = G $ [closed]

I'v found this proposition. If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $. Why this is true ?
1
vote
2answers
176 views

Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
0
votes
1answer
100 views

Normal subgroups of finite solvable groups

Let $G$ be a finite solvable group, $N$ a nontrivial abelian normal subgroup of prime exponent $p$. Let $Q$ be a $p$-Sylow subgroup of $G$ containing $N$. Is it possible that the normal core of ...
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0answers
64 views

Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
2
votes
3answers
75 views

Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why ...
6
votes
1answer
89 views

The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$

I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
2
votes
2answers
112 views

The sum of the orders of all elements of a group G

Let $Z$ be a finite group and denote $k(Z)$ the sum of the orders of all the elements of the group $Z$. I have to determine min $k(Z)$ and max $k(Z)$ when $G$ goes through the set of the abelians ...
0
votes
2answers
70 views

Injective group homomorphism between $D_6$ and $S_5$

Is there an injective group homomorphism between $D_6$ and $S_5$, where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group?
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1answer
72 views

Compute factor group $\dfrac{\mathbb{Z}_4 \times \mathbb{Z}_6}{\langle(2,3)\rangle}$ - Fraleigh p. 147 Example 15.11

(1.) Why's there a 'great temptation' to set $2 \bmod 4$ and $3 \bmod 6$ to 0? (2.) Why are you authorized to set $2 \bmod 4$ and $3 \bmod 6$ to 0? $2 \bmod 4 \neq 0$ and $3 \bmod 6 \neq 0$, hence ...
0
votes
1answer
41 views

common element of subgroups of $p-$group $G$ and generator set of $G$

Consider a $p-$group $G$ and a set $S$ which generates $G$ and $|S|>5$. (I can consider the case that $S$ is minimal) consider an arbitrary non trivial subgroup $H$ of $G$. It s clear that there ...
5
votes
2answers
214 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
1
vote
1answer
60 views

What is $\mathbb{Z}_2 \times \mathbb{Z}_4$ isomorphic to - Fraleigh p. 112 Exercises 11.32e

(e). p. 4 of PDF - $\mathbb{Z}_2 \oplus \mathbb{Z}_4 \not\simeq \mathbb{Z}_8$. Another solution (1.) Why is $\mathbb{Z}_2 \oplus \mathbb{Z}_4$ not cyclic? Is it because of $ \gcd(2, 4) = 2 \neq 1 ...
2
votes
1answer
57 views

Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...