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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Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
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Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why ...
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The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$

I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
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The sum of the orders of all elements of a group G

Let $Z$ be a finite group and denote $k(Z)$ the sum of the orders of all the elements of the group $Z$. I have to determine min $k(Z)$ and max $k(Z)$ when $G$ goes through the set of the abelians ...
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70 views

Injective group homomorphism between $D_6$ and $S_5$

Is there an injective group homomorphism between $D_6$ and $S_5$, where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group?
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Compute factor group $\dfrac{\mathbb{Z}_4 \times \mathbb{Z}_6}{\langle(2,3)\rangle}$ - Fraleigh p. 147 Example 15.11

(1.) Why's there a 'great temptation' to set $2 \bmod 4$ and $3 \bmod 6$ to 0? (2.) Why are you authorized to set $2 \bmod 4$ and $3 \bmod 6$ to 0? $2 \bmod 4 \neq 0$ and $3 \bmod 6 \neq 0$, hence ...
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1answer
41 views

common element of subgroups of $p-$group $G$ and generator set of $G$

Consider a $p-$group $G$ and a set $S$ which generates $G$ and $|S|>5$. (I can consider the case that $S$ is minimal) consider an arbitrary non trivial subgroup $H$ of $G$. It s clear that there ...
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214 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
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1answer
60 views

What is $\mathbb{Z}_2 \times \mathbb{Z}_4$ isomorphic to - Fraleigh p. 112 Exercises 11.32e

(e). p. 4 of PDF - $\mathbb{Z}_2 \oplus \mathbb{Z}_4 \not\simeq \mathbb{Z}_8$. Another solution (1.) Why is $\mathbb{Z}_2 \oplus \mathbb{Z}_4$ not cyclic? Is it because of $ \gcd(2, 4) = 2 \neq 1 ...
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57 views

Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...
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39 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
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1answer
38 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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1answer
67 views

Intersection of $p$-subgroup normalizer

Let $Q \leq S$ with $S$ a Sylow $p$-subgroup of $G$. I am interested in conditions that guarantee $$R_Q = \bigcap\left\{ N_{S^g}(Q) : g \in N_G(Q) \right\}$$ is equal to $Q$. For instance $Q=S$ ...
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140 views

Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
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Non-abelian finite groups with exactly $n$ normal subgroups.

Let $\mathfrak{N}$ be the class of all non-abelian finite groups and define $\nu: \mathfrak{N} \rightarrow \mathbb{N}_{\gt 1}$ by $\nu(G)=|\{{1} \leq N \leq G: N$ normal in $G\}|$. Is the map $\nu$ ...
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1answer
50 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
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308 views

Nonisomorphic groups of order 12.

I'm trying to find 4 groups of order 12, none of which are isomorphic to each other. Should i be trying external direct products? So far i have $A_4, \mathbb Z_{12},\,$ and $\,\mathbb Z_6\times ...
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2answers
175 views

Homomorphisms from $D_4$ to $S_3$.

Find all homomorphisms from $D_4$ to $S_3$. We have $D_4 = \{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$ (where $r^4 = e = s^2$) and $S_3 = \{e,(12),(13),(23),(123),(132)\} = \langle (12) (13) \rangle$. Let ...
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1answer
487 views

Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems

I'm showing that any group of order $63$ has an element of order $3$, and can only use Lagrange's theorem not Cauchy's or Sylow's. I got it reduced to a case of having $62$ elements of order $7$ but ...
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1answer
63 views

A group of odd order has no non-identity elements which are conjugate to their inverse.

I want some verification for my proof to a homework problem. (Is it correct? Is there a simpler way to do this?) Let $G$ be a finite group of odd order and suppose there is an element $g$ that is ...
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658 views

What is the intersection of all Sylow $p$-subgroup's normalizer?

Intersection of all Sylow $p$-subgroups is generally denoted by $O_p(G)$ and it is one of the well studied topics in group theory as there are many theorems related to this. Let $R$ be ...
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101 views

Is the group $G$ cyclic?

Assume that $G$ is a finite group such that for any positive integer $n$ dividing $|G|$, $G$ has one and only one subgroup $H$ with $|H|=n$. Is $G$ cyclic?
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90 views

Prove that $N \cap K$ is a normal subgroup in $K$.

Let $N$ and $K$ be subgroup of a group $G$. If $N$ is normal in $G$, prove that $N \cap K$ is a normal subgroup of $K$. Since $N$ is normal in $G$, we have $Ng = gN$ for some $g \in G$. Also ...
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Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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1answer
46 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
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2answers
76 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
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1answer
385 views

Every finite group of order more than two has a nontrivial automorphism [duplicate]

I want to prove that every finite group $G$ of order more than 2 has a nontrivial automorphism. I've seen this question answered on this site for infinite groups, but the proofs given use the fact ...
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318 views

How many different elements can we obtain by multiplying all element in a group?

Let $G$ be a finite group. How many different elements can we obtain by multiplying all element in a group? Of course, if $G$ is abelian the answer is one but when G is non-abelian, changing the ...
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1answer
78 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
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1answer
295 views

Sylow p-subgroups, normal subgroups and the center subgroup

Let: $G$ be a finite group. $p$ be a prime number. $P$ be a Sylow-p subgroup of $G$. If $p\mid o(G)$ and for every $(a,b)\in G$, $(ab)^p=a^pb^p$, please help me prove the following: (1) ...
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1answer
385 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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1answer
79 views

Are these two inclusions of finite groups, equivalent?

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a group morphism, and ...
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86 views

How many different group homomorphisms from $\mathbb{Z}/\mathbb{3Z}$ into $SL_{n}(7)$ are there?

I have the following question: How many different group homomorphisms from $\mathbb{Z}/\mathbb{3Z}$ into $SL_{n}(7)$ are there? I think that the answer is the number of elements of order $3$ in ...
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60 views

How to list all permutations of $S_n$ for any given value of $n$.

In writing out a symmetric subgroup $S_n$ with some given $n$-value, how would I go about doing this? For example, allow me to attempt with $n=3$: $S_3 = \{ (1 3), (2 3), (1 2), (1 2 3), (1 3 2), (2 ...
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Showing $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_4$ have a different number of subgroups of order $2$.

Looking at this example, can someone explain to me what is $_1$, $H_2$, $H_3$, and how they came about it?
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1answer
60 views

Embedding symmetric groups into general linear groups

Note that any finite abelian group $\Gamma$ can be embedded into $\operatorname{GL}(1,A)$ for some commutative ring $A$. Indeed, let $A=\mathbf Z[\Gamma]$ and $\Gamma\hookrightarrow ...
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48 views

Conjugate element

Let $G$ be a group of order $2014$. Let $\theta\in G$ such that $|\theta|=19$ and $\alpha\in G$ such that $|\alpha|=2$. Show that $\alpha\theta\alpha=\theta^{\pm1}$. Since the order of $\alpha$ is 2 I ...
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65 views

general linear groups and definitions

We have two groups, one of them is automorphisms group of a vector space over GF(2) and another one is the direct product of two automorphism group (they are also over GF(2)). Also, via some ...
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1answer
115 views

Proving that there exists an element of order $p^2$ in a finite abelian group

I've been stuck on this problem for a while now. Let $G$ be finite and abelian. Suppose $\exists x \in G$ such that $x$ has non-square-free order, i.e., $|x| = p^km$ with $p$ prime and ...
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103 views

Estimate the Number of Conjugacy Classes of $G$

This is a series of questions in my book unanswered. Let $c(G)$ be the number of conjugacy classes in $G$. Define $\bar{c}(G):=\frac{c(G)}{|G|}$. Now we estimate the $\bar{c}(G)$ of a non-abelien $G$. ...
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Identity based encryption

I am implementing ID based encryption in c# right now i am having problem at the following mathematics $$H_1: \{0,1 \}^n \times \{0,1 \}^n \to \mathbb{Z}^*_p, \text{ what does this expression ...
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1answer
241 views

Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
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1answer
55 views

Abstract Algebra: Index of Subgroups

Here's the problem I'm working on: Prove: Suppose $H$ has index $p$ and $K$ has index $q$, where $p$ and $q$ are distinct primes. Then the index of $H \cap K$ is a multiple of $pq$. (Plus: do you ...
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From linear invariants of group to general ones

There is a lot of information about classical/linear invariants of finite groups. But does it lead to general invariants of group (for example, when we consider some action of our group on finite ...
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1answer
28 views

Understanding semi-direct product construction

I am a student studying semi-direct products for the first time, and have this question:say $G = N \rtimes H$, where $N$ is normal and $H$ is another subgroup that "acts" on $N$. The quotient $G/N ...
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158 views

Why is studying maximal subgroups useful?

When looking at finite group theory research, it seems to me that a lot of energy is devoted to determining the maximal subgroups of certain classes of groups. For example, the O'Nan Scott theorem ...
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72 views

Residually finite group with finitely many conjugacy classes of elements of finite order

Let $G$ be a residually finite group. Show that if $G$ has finitely many conjugacy classes of elements of finite order then $G$ has a torsion free finite index subgroup. Not sure how to get started ...
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68 views

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$?

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$? I try to understand how to solve it but I don't finding a way... I'll be glad if you help me with this... Thank you!
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128 views

Calculating sylow subgroups of some concrete groups

Question is to : exhibit all sylow $3$ - subgroups of $S_4$ What i have done so far is : Number of elements in symmetric group is $|S_4|=4.3.2=2^3.3$ number of elements of order $3$ in $S_4$ ...
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63 views

Finite groups with no common prime factor of their orders

Let $G$ and $H$ be finite groups s.t. their orders have no common prime factor, and let $\phi: G\rightarrow H$ be a homomorphism. I want to show that $\phi(g)=e \space \forall g \in G$ where $e$ is ...