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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7
votes
3answers
32 views

Proving that $D_{12}\cong S_3 \times C_2$

Prove that $D_{12}\cong S_3 \times C_2$. I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how ...
-1
votes
0answers
24 views

Cyclic subgroups of $\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$

$\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$ has 24 elements of order 10. Why each cyclic subgroup of order 10 has four elements of order 10 ?
2
votes
2answers
29 views

Group Theory and Lagrange's Theorem: coprime subgroups.

Let $G_1$ and $G_2$ be finite groups, and let $K≤G_1 \times G_2$. Let $H_1 = \{ g \in G_1 : (g,e) \in K\}$ and $H_2 = \{g \in G_2 : (e,g) \in K\}$ and suppose $|G_1|$ and $|G_2|$ are coprime. Then ...
0
votes
1answer
38 views

Show that this group is nilpotent.

Let $G$ be a finite solvable group whose order is divisible by at least three distinct primes. If every Hall $p'$-subgroup of $G$ is nilpotent, show that $G$ is nilpotent. I feel like the best ...
0
votes
0answers
31 views

Need an example

Let $p$ be a prime number. I need an example of finite group $G$ generated by the elements of order $p^n$ ($n\in \mathbb N$) , which contains a normal subgroup $H$ that is not generated by the ...
3
votes
0answers
39 views

I can't complete a proof of Plancherel.

I have been trying to prove the following theorem: Let $G$ to be a finite group and $u = \sum_{s\in G}u(s)s$, $v = \sum_{s\in G} v(s)s$ be elements of $\mathbb{C}[G]$. Then $$\frac{1}{|G|} ...
-1
votes
0answers
22 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$

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. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
3
votes
1answer
57 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
-2
votes
0answers
19 views

Let $N$ be a minimal normal subgroup of $G$, then $G/N$ is supersoluble? [on hold]

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. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
2
votes
1answer
29 views

If $G$ is finite group that supersoluble then $G$ satisfy the maximal permutizer condition?

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. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
-1
votes
0answers
24 views

What is the permutizer of the Sylow 3 subgroup in $S_4$ ? [on hold]

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. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
0answers
18 views

can say every group that satisfy in maximal permutizer condition then satisfy then permutizer condition

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. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
0
votes
1answer
17 views

How to find subgroup centralizer?

Having found that a group G has a normal Sylow 2-subgroup P, how do I find $C_P(g_i)$, where $g_i$ is a conjugacy class representative? I have the character table, and have previously found ...
0
votes
1answer
22 views

What is the group $C_2^4$?

I'm trying to do a problem which asks me to show that a certain group is isomorphic to $C_2^4$. What is this group?
2
votes
1answer
32 views

Subgroups of every order dividing the order of the group imply the group is abelian?

Let $G$ be a finite group, denote $|G|=n$. I know about Cauchy theorem which states that if for a prime $p$: $p|n$ then there is $H\leq G$ with $|H|=p$. I also know that an abelian group $G$ have a ...
2
votes
2answers
30 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
3
votes
0answers
49 views

Normal subgroup of General linear group

What is the list of all normal subgroups of general linear group $GL_n(q)$? (n*n invertible matrix on finite field with $q$ elements) It is well known $SL_n(q)$ and subgroups of $Z(GL_n(q))$ are ...
0
votes
0answers
18 views

Coercing elements into a set in MAGMA

Suppose I had a permutation group $G$ for example, that corresponded to the intersection of some elements of a list $H$ where the elements are groups. Then, if I wanted to create a set or some ...
0
votes
2answers
14 views

Two subgroups whose orders are greater than square root of the group order have no trivial intersection

Two subgroups whose orders are greater than square root of the group order have no trivial intersection I cannot come up with a critical idea.
1
vote
1answer
45 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
0
votes
0answers
35 views

Fundamental Domain of ${\mathbb Z}^2$ to ${\mathbb R}^2$

Find a fundamental domain for the action of $\mathbb Z^2$ on $\mathbb R^2$ by translation A fundamental domain is the nodes $(0,0),(1,0),(0,1)$ and the edges which connect them Is there a better way ...
0
votes
0answers
29 views

subsets of $\mathbb{Z}_2^{p}$ up to permutation equivalence

Let $\mathbb{Z}_2:= \mathbb{Z}/2\mathbb{Z}=\{0,1\}$. Let $p$ be a prime integer. We use $$\mathbb{Z}_2^{p}:= \mathbb{Z}_2\times \mathbb{Z}_2 \cdots \times\mathbb{Z}_2\qquad (p-times).$$ i.e., each ...
1
vote
1answer
39 views

Non-trivial group homomorphism from an infinite group to a finite group

Let $G$ be a topological group the underlying set of which is infinite (e.g., $(\mathbb{R}\,;+)$ or $(\mathbb{Z}\,;+)$), and let $H$ be a topological group the underlying set of which is finite (e.g., ...
0
votes
1answer
18 views

Uniquely $p$-divisible group-Reference Request.

Define a group $G$ to be uniquely $p$-divisible if for all $x\in G$, there is a unique $y\in G$ such that $x=y^p$. Can someone kindly provide references where this class of groups is studied? Of ...
0
votes
1answer
19 views

If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

If $G$ is a finite group such that their is a normal subgroup $H$ of $G$ such that $G/H \cong V_4$ , where $V_4$ is the Klein four group , then is it true that $G$ can be written as a union of three ...
1
vote
0answers
36 views

An Abstract Characterization of $S_5$ using involutions and their centralizers

This is essentially an exercise from Jacobson's Basic Algebra I. (p.83, ex.10) I've managed to solve all the other part of the proof, except (vi) and (x). I've been thinking about this all day, but ...
1
vote
1answer
35 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [closed]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
0
votes
0answers
24 views

Conjugacy classes of solvable groups [closed]

If $A$ be a subset of solvable group $G$, let $k_G(A)$ be the number of $G$- conjugacy classes contained in $A$. Also let $N$ be a normal subgroup of $G$ of odd order, $\frac{G}{N}$ be an abelian ...
1
vote
1answer
31 views

question on lemma in Bushnell and Henniart, irreducible components of a particular induced representation

I have a question on a lemma that appears in the book "The Local Langlands Conjecture for GL(2)" by Bushnell and Henniart. The setting is as follows: we let $G = GL_2(k)$ where $k$ denotes a finite ...
0
votes
1answer
18 views

Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
0
votes
0answers
32 views

Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
5
votes
2answers
58 views

If $|G|=p^n$, then $p^2 \le |G : G^\prime|$. [closed]

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
3
votes
1answer
55 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
2
votes
1answer
33 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
2
votes
1answer
49 views

If two subsets $S,T\subseteq G$ have sum of cardinalities greater than $|G|$, then $S+T=G$ [duplicate]

Let $S$ and $T$ are two subset of a finite group $(G,+)$ so that $|S|+|T|>|G|$, then Prove that $S+T=G$, where $S+T=\{s+t:s\in S ,t\in T\}$ My effort: It is clear that $S+T\subseteq G$ as ...
0
votes
0answers
37 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
-3
votes
1answer
33 views

Subgroup of a cyclic finite group [closed]

Let $G$ be a cyclic group of order $n$ and let $m$ be a positive integer dividing $n$. Show that $G$ contains one and only subgroup of order $m$. I have started the proof by stating the smallest ...
0
votes
2answers
19 views

Sylow counting - classifying groups of order 15

let $G$ be a group of order $15$. this is the argument I was given: We have $15 = 3\times 5$ so we start with $p = 5$ We have by Sylow's theorem that $N_5 = 1 \mod 5$ so $N_5 = 1$ or $N_5 \geq 6$. ...
-2
votes
4answers
58 views

Prove the only homomorphism between groups with coprime orders is trivial. [closed]

Deduce that if $\gcd(|G|, |H|)=1$ then the only homomorphism $\phi : G \rightarrow H$ is trivial.
2
votes
0answers
39 views

Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
1
vote
2answers
31 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
5
votes
1answer
40 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
2
votes
1answer
43 views

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$
1
vote
1answer
51 views

If $\left<3\right> \cong \mathbb Z_2$ and $\left<2\right> \cong \mathbb Z_3$, why isn't $\left<3\right>\left<2\right> \cong \mathbb Z_2\mathbb Z_3$

$\left|\mathbb Z_{6}\right|=6=2 \cdot 3$. Since $\mathbb Z_{6}$ is abelian, all subgroups are normal and thus its Sylow subgroups are unique. $\left<3\right>$ is the $2$-Sylow, and ...
3
votes
1answer
59 views

Working with finitely presented groups in GAP

This is more of a question specifically about how GAP handles calculations with finitely presented groups rather than about group theory. I have several finite group presentations that I would like ...
1
vote
1answer
20 views

Multiple Group Representations using Cayley's Thm

I know that an abstract group can be made isomorphic to a subgroup of a symmetric group, by using a Cayley table for that abstract group. However, what is a technique for getting another permutation ...
2
votes
3answers
77 views

Why isn't $\mathbb Z_9 \cong \mathbb Z_3 \times \mathbb Z_3$ by the fundamental theorem of finite abelian groups?

I was reading the answer to this question: Explicit descriptions of groups of order 45 and the accepted answer says the Sylow $3$-subgroup is either isomorphic to $\mathbb Z_9$ or $\mathbb Z_3 \times ...
3
votes
1answer
56 views

Various Intersections of Sylow p-subgroups.

I was told yesterday that in a system of Sylow $p$-subgroups of a finite group $G$, if, $\{S_1,S_2, \cdots, S_n\}$ make up the system, it can happen that, say, the intersection of $S_1$ and $S_2$ has ...
5
votes
2answers
104 views

Algorithm design for enumerating pairs of noncommuting elements up to conjugacy

I am trying to write some Magma code that, given a group $G$, returns a list of pairs $(x,y)$ with $x,y\in G$ such that $[x,y]\neq 1$ and such that every pair $(z,w)$ in the group with $[z,w]\neq 1$ ...
4
votes
2answers
66 views

Elements of order three in $GL_3(2)$

How do I go about finding elements of order 3 in $GL_3(2)$? I'm currently trying to show that the automorphism group of a Klein 4-group induced by conjugation in $GL_3(2)$ is isomorphic to $S_3$ so am ...