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
2answers
58 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: * ...
1
vote
1answer
39 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 ...
1
vote
1answer
85 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 ...
4
votes
1answer
42 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
41 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 $\Bbb F$?
4
votes
4answers
77 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 ...
3
votes
1answer
80 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 ...
-1
votes
1answer
57 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
170 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
vote
1answer
59 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 ...
1
vote
0answers
63 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
43 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
36 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
27 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, ...
4
votes
2answers
72 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
76 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 ...
3
votes
0answers
64 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
82 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 ...
3
votes
2answers
113 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
23 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
33 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
190 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
67 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 ...
0
votes
1answer
58 views

If no element in G has order n, can I say no subgroup in G has order n? [closed]

Suppose $G$ is a finite group. If no element in $G$ has order $n,$ can I say no subgroup in $G$ has order $n$? Thanks
2
votes
0answers
44 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
87 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
38 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, ...
3
votes
1answer
120 views

Qualifying Exam Question On Elementary Group Thoery

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
65 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
43 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 ...
6
votes
1answer
118 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
46 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 ...
8
votes
3answers
221 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 ...
4
votes
1answer
138 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
27 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
39 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
120 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
50 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
27 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
54 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
178 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
39 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
31 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
30 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
34 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
26 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
32 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
0answers
78 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
17 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
76 views

Order of a specific group

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