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
votes
0answers
18 views

Show that the order of the class of $p+1$ in $\left( \mathbb{Z}/p^{\alpha}\mathbb{Z}\right)^{*}$ is $p^{\alpha-1}$

I tried to do that: $$(1+x)^p=1+px+\binom{p}{2}x^2+\dots+\binom{p}{p-1}x^{p-1}+x^p$$ So if $x=0 \quad mod(p^k)$ then $(1+x)^p=1 \quad mod(p^{k+1})$ Now I'm trying to deduce that ...
3
votes
1answer
89 views

Order of a specific group

Is it true that the order of this group is $14$ (because of $7\cdot2$)? $$\langle S, T\mid S^7 = (S^4T)^4 = (ST)^3 = T^2 = 1\rangle$$
3
votes
1answer
33 views

Order of $\left(\mathbb{Z}/10\mathbb{Z}\right)^{*}$ and others groups

I already know that $\left(\mathbb{Z}/10\mathbb{Z}\right)^{*}=\left(\mathbb{Z}/2\mathbb{Z}\right)^{*}\times\left(\mathbb{Z}/5\mathbb{Z}\right)^{*}$ Theses groups have order 1 and 4 so the group is ...
2
votes
2answers
72 views

Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

I'm trying to determinate how many morphisms of groupes exist between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ for $n,m\in\mathbb{N}$. I know a morphism is determinated by the image of ...
3
votes
1answer
184 views

an order of automorphism group of finite abelian group

This is problem of Rotman's Exercise 7.9(i). If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order. How can I approach to this problem? Could you suggest ...
5
votes
1answer
75 views

Automorphisms of spin groups over finite fields, even dimension

I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of ...
2
votes
1answer
220 views

If $|A| > \frac{|G|}{2} $ then $AA = G $ [closed]

I'v found this proposition. If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $. Why this is true ?
1
vote
2answers
219 views

Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
0
votes
1answer
123 views

Normal subgroups of finite solvable groups

Let $G$ be a finite solvable group, $N$ a nontrivial abelian normal subgroup of prime exponent $p$. Let $Q$ be a $p$-Sylow subgroup of $G$ containing $N$. Is it possible that the normal core of ...
1
vote
0answers
68 views

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 $n$ ...
2
votes
3answers
104 views

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 ...
6
votes
1answer
101 views

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.
2
votes
2answers
158 views

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 ...
0
votes
2answers
77 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?
1
vote
1answer
80 views

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 ...
0
votes
1answer
45 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 ...
5
votes
2answers
227 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 ...
1
vote
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 ...
2
votes
1answer
66 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 ...
1
vote
1answer
44 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 ...
2
votes
1answer
42 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 ...
2
votes
1answer
76 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 $Q=S$ ...
3
votes
1answer
155 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 ...
6
votes
2answers
96 views

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 $\nu$ ...
1
vote
1answer
51 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 ...
1
vote
3answers
566 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 ...
1
vote
2answers
180 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 ...
3
votes
1answer
694 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 ...
1
vote
1answer
73 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 ...
10
votes
3answers
814 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 ...
0
votes
2answers
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?
1
vote
2answers
100 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 ...
1
vote
0answers
89 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
0
votes
1answer
51 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 ...
4
votes
2answers
84 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 ...
4
votes
1answer
713 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 ...
15
votes
2answers
333 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 ...
0
votes
1answer
82 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 ...
2
votes
1answer
392 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) ...
1
vote
1answer
579 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 ...
1
vote
1answer
82 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 ...
3
votes
1answer
94 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 ...
2
votes
2answers
62 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 ...
-1
votes
1answer
40 views

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?
2
votes
1answer
81 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 ...
4
votes
1answer
53 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 ...
0
votes
1answer
76 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 ...
1
vote
1answer
154 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 ...
5
votes
1answer
124 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 $G$. ...
1
vote
0answers
54 views

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 ...