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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2
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1answer
76 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 ...
3
votes
1answer
59 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
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0answers
62 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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1answer
49 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
48 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 $N_5 \geq 6$. ...
10
votes
2answers
166 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
52 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
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1answer
106 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
50 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
56 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
106 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
23 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
133 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
85 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
228 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
67 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 ...
0
votes
4answers
47 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 ...
0
votes
0answers
32 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 ...
0
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1answer
46 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 ...
2
votes
2answers
102 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
122 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?
0
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1answer
55 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
votes
1answer
58 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 ...
0
votes
1answer
53 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
votes
1answer
41 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: ...
2
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1answer
44 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 & ...
5
votes
3answers
109 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
65 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
40 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, ...
0
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1answer
59 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. ...
2
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2answers
43 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 ...
3
votes
1answer
198 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
23 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
votes
2answers
61 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
37 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
62 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 $\mathbb{C}G$. ...
0
votes
1answer
78 views

$GL_3(\mathbb{F}_2)$ is simple

Character table of $GL_3(\mathbb{F}_2)$. \begin{array}{|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 7A & 7B \\ \hline \chi_1 & 1 & 1 & 1 & 1 & 1 & ...
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0answers
46 views

What does it mean for a subgroup to be self-centralising in terms of group extensions

In texts on group theory I read about subgroups $U \le G$ which fulfill the property $$ C_G(U) \le U $$ (this is called self-centralising, for example the Fitting subgroup in solvable groups fulfills ...
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1answer
42 views

End of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Near the end of the proof of Burnsides $p^aq^b$ Theorem, we want to prove the following If $\rho:G ...
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0answers
45 views

Middle bit of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Part of the proof needed to prove the above theorem is where you prove that: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
1
vote
1answer
42 views

Number of orthonormal sets of vectors $\leq$ dim of the vector space

Theorem. If $V,W$ are arbitrary representations of $G$, say $$V=V_1^{a_1}\oplus \dots \oplus V_k^{a_k} $$ $$W=V_1^{b_1}\oplus \dots \oplus V_k^{b_k} $$ Then, $$\langle \chi_V,\chi_W \rangle ...
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2answers
67 views

Start of proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Proof. Enough to prove that no non-abelian simple groups have order $p^aq^b$. [Then break $G$ into simple pieces ...
0
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1answer
123 views

Left and right multiplication for Cayley graphs

Correct me if I am wrong but I have written down the following: Let $X$ be a finite group, with subset $S$ and corresponding Cayley graph $G$. The edge set for a Cayley graph is defined such that ...
2
votes
1answer
85 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
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vote
2answers
42 views

Generator of group, find the inverse, solve equation

Given the prime number $p=101$ Find a generator of the group $\mathbb{Z}_p^{\star}$. How many generators of $\mathbb{Z}_p^{\star}$ are there? Find $5$ generators. Find the inverse of $\overline{83}$ ...
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0answers
47 views

Subgroups and Direct Products of $\pi$-closed groups are also $\pi$-closed.

Let $K$ be a class of finite groups, then this class is called closed iff i) homomorphic images of groups in $\mathcal K$ lie in $\mathcal K$, ii) subgroups of groups from $\mathcal K$ lie in ...
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1answer
52 views

Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
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2answers
93 views

Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$

Here is the full question : Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$ onto a subfield of the algebraic closure of ...
2
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1answer
87 views

Groups of Order $210$

By the "$2n$-test", proving that a group of order $210$ cannot be simple. Is there another way to prove this? Would you use Sylow Theory?
3
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1answer
39 views

Is there any generalization of Frobenius group?

Let $G$ act on $\Omega$ transitively and $\chi(g)$ is equal to number of the elements fixed by $g$. If $\chi(g)\leq 1$ for all $g\in G\setminus\{e\}$ then $G$ is a Frobenius group. There are many ...