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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1answer
15 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 ...
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0answers
31 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 ...
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1answer
35 views

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

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 $G$.
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0answers
23 views

Conjugacy classes of solvable groups [on hold]

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 ...
3
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0answers
28 views

How do you define a Fourier transform in the non-commutative case?

I have been trying to figure out how Fourier Transforms over a non-commutative group by starting with some relatively simple finite groups I've been getting fairly confused. The paper found here by ...
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1answer
28 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 ...
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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?
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0answers
27 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
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2answers
56 views

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

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 $n \ge 2$.
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1answer
52 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 $x$, ...
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1answer
31 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
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1answer
48 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 ...
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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 ...
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0answers
27 views

Complement of a Hall subgroup [on hold]

Let $G$ be a finite group and let $F(G)$ be the Fitting subgroup of $G$. Show that if $H$ is a Hall $\pi$-subgroup of $G$ and $H/G$ is complemented in $G/F(G)$, then $H$ is complemented in $G$. (see ...
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1answer
32 views

Subgroup of a cyclic finite group [on hold]

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 ...
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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 $N_5 \geq 6$. ...
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4answers
57 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.
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0answers
38 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 $|G| = 8n$, ...
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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 ...
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0answers
31 views

What do we know about a group when we know it's order? [closed]

Specifically, if a Group has order $rs$, what can we automatically know about this group?
5
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1answer
39 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
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1answer
42 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 $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$
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1answer
49 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
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1answer
38 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
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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
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3answers
75 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 ...
2
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1answer
45 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, {S1,S2,..Sn} make up the system, it can happen that, say, the intersection of S1 and S2 has order p^k, while the ...
5
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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
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2answers
65 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 ...
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4answers
42 views

Proof involving Lagrange's Theorem

Let $a$ and $b$ be nonidentity elements of different orders in a group $G$ of order $155$. Prove that the only subgroup of $G$ that contains $a$ and $b$ is $G$ itself. What I have so far: We know ...
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0answers
25 views

Geometric understanding of why $D_1\cong \mathbb{Z}_2$

From what I understand, $P_2$ is the line joining $(-1,0)$ to $(1,0)$ and then $P_1$ is the point $(1,0)$. This is due to defining $P_n$ (the regular polygon with # of points $n$) to be formed by the ...
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1answer
32 views

Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$

Let $G=S_3$ and $H=\langle (12) \rangle < G$. Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$. I have computed the character using the induction ...
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2answers
87 views

Conjugacy classes of the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$

Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. This group does not appear to be easy to work with! Does anyone know what this group is called? I am trying to find its ...
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2answers
47 views

What would the notation G/H mean in terms of groups and subgroups?

What would G/H mean in terms of subgroups? Would it most likely mean The compliment group of H in G?
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1answer
49 views

Using generators to write representations

Let $G=D_{12}=\{a,b\mid a^6=b^2=1, bab=a^{-1}\}$. Also let $A=\begin{pmatrix} e^{i\pi/3} & 0 \\ 0 & e^{-i\pi/3} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
2
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1answer
45 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
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1answer
38 views

Only one cancellation law? Then $G$ may not be a group…

Suppose that the following result is known: "Let $G$ be a finite set, closed with respect to an associative product and that both of the cancellation laws are valid. Then $G$ is a group with ...
2
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1answer
24 views

Prove that the inner product is the number of orbits of $G$ on $X \times Y$.

Let $X,Y$ be $G$-sets and $\mathbb{C}[X], \mathbb{C}[Y]$ the corresponding permutation representations. Prove that the inner product is the number of orbits of $G$ on $X \times Y$. Ive tried: ...
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1answer
28 views

What is the process behind finding a Cayley permutation representation.

For example, let's find the Cayley permutation representation of $\mathcal D_3$ in $S_6$. $\mathcal D_3 = \left<r,s \mid r^3=s^2=1, rs=sr^{-1}\right>$. Write, \begin{pmatrix} 1 & 2 & ...
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3answers
92 views

Let $G$ be a finite simple group. Suppose that $A, B < G$, $G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $?

Let $G$ be a finite simple group. Suppose that $A$ and $B$ are proper subgroups of $G$, $ G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $ ? I checked it with some examples and it ...
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2answers
51 views

Show that $\rho$ must be 2-dimensional

Let $G=D_8=\langle g,h |g^4=h^2=1, hgh=g^{-1} \rangle$. One can show that $G$ has $4$ $1$-dimensional representations. From first principles (no character theory). Suppose $\rho$ is an ...
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0answers
16 views

Question about exponents of groups

Okay, I'm trying to understand exponents of groups. I will start with the Set Z_3, where Z_3 is the integers mod3 under addition. Now, I want to set out to find the exponent of this group, but ...
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1answer
43 views

Finding a group that is not monomial

Definition. A group is called monomial if every representation of $G$ is induced from 1-dimensional representations of some subgroup of $G$. Question Give an example of a group that is not monomial. ...
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2answers
34 views

Finding the factor of the derived subgroup of non-abelian group of order 12

Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. The derived subgroup is $G'=\{e,a^2,a^4\}$ I believe. So I am trying the find $G/G'$. Now I know that $|G/G'|=4$ so it ...
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0answers
57 views

Number of group homomorphisms between two finite groups

I am confused between the Answer of this Question 1 and the Answer of this Question 2. In answer of the 1st question groups should must be Abelian whether in the answer of 2nd question there are no ...
2
votes
0answers
32 views

Proof of the Frobenius Schur indicator

I am trying to prove the Frobenius-Schur indicator for $\chi$ irreducible character. \begin{equation} i_{\chi} = \begin{cases} 0, & \text{if $\chi$ is not real valued} \\ \pm1, & ...
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0answers
12 views

Components and Centralisers of Involution. A wrong argument?

A component $K$ in a finite $G$ is a subnormal subgroup which is quasisimple, i.e. perfect and $K/Z(K)$ is simple. Obviously, when $K$ is a component of $G$ and $U\le G$, then $K$ is also a component ...
6
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2answers
53 views

The minimal group with Fitting length three

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. If every proper subgroup of $G$ and every non-trivial quatient of $G$ has ...
0
votes
1answer
26 views

Sufficient Conditions for the Commutator Subgroup to be a Component

A group $K \ne 1$ is quasisimple if $K$ is perfect and $K/Z(K)$ is simple. For every subnormal subgroup $N$ of a quasisimple group $K$ either $$ N \le Z(K) \quad \mbox{ or } \quad N = K. $$ A ...
4
votes
2answers
48 views

Product of class sums

Let $C_i$ be the conjugacy classes of a finite group $G$. Consider the class sums $z_i=\sum_{g\in C_i} g$. It is well known that ${z_i}$ form a basis of the center of the group algebra $\mathbb{C}G$. ...