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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Is this formulation of Plancherel's Theorem correct?

Let $G$ be a finite group, $u,v\in \mathbb{C}[G]$ where $u = \sum u(g)g, v = \sum v(g)g$, and $\rho_{i}: G\rightarrow GL(W_{i})$ be irreducible representations and $\tilde{\rho_{i}}: ...
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1answer
30 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., ...
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0answers
22 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 ...
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0answers
12 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 ...
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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 $[z,w]\neq 1$ ...
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3answers
2k views

How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? [duplicate]

How do I find the number of group homomorphisms from the symmetric group $S_3$ to $\mathbb{Z}/6\mathbb{Z}$?
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1answer
16 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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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 ...
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0answers
33 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
41 views

left and right multiplication for Cayley graphs

Correct me if i am wrong but i have written down the following: If $X$ is a finite group, with subset $S$ and corresponding Cayley graph $G$ The edge set for a Cayley graph is defined such that two ...
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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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1answer
29 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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24 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 ...
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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
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
30 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 ...
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3answers
79 views

If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.

Let $G$ be a finite group. Suppose that $A$ and $B$ are to subsets of $G$. If $|G|<|A|+|B|$ prove that $$G=AB.$$
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1answer
53 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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0answers
170 views

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
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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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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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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 ...
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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 [closed]

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 [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 ...
4
votes
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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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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1answer
115 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
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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
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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4answers
2k views

There exists only two groups of order $p^2$ up to isomoprhism.

I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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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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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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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1answer
46 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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0answers
26 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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2answers
145 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
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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 $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
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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
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 ...
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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: ...
4
votes
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 ...