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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3
votes
1answer
68 views

automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
2
votes
1answer
47 views

cyclic group definiton

i would like to clarify some things related to group theory,for example let us consider two group,one with modulo $3$ and second with modulo $2$,so we have $G_3=(0,1,2)$ and $h_2=(0,1)$ now if i ...
3
votes
2answers
69 views

If a finite group has all $p$-complements, is it always solvable?

On reading the question Subgroups of Prime Power Index I immediately thought "if the group had a $p$-complement for each prime $p$ then it would be solvable". But then I realized that the argument I ...
2
votes
2answers
72 views

Subgroup of a (free) group.

Let $F$ be a group, generated by $x_1,...,x_m$ and $H$ be its subgroup such that $|F:H|=n < \infty$. How to prove that $H$ can be generated by $n(m-1)+1$ elements?
3
votes
1answer
71 views

Suzuki exceptional characters with $\epsilon = -1$

I have a question about Suzuki's theory of exceptional characters of finite groups. If you are familiar with this theory, then I'm just asking: can we always choose $\epsilon=1$? If not, here is a ...
1
vote
1answer
123 views

show that the following three statements are equivalent

Let $\Phi$ be an irreducible complex representation of the group $S_n$ and $\Phi'(\sigma)=\Phi(\sigma) \operatorname{sgn}(\sigma).$ $(\sigma \in S_n).$ Prove that 1) $\Phi'$ is ...
1
vote
1answer
52 views

Under what conditions should a sub-group of a direct sum, itself be a direct sum?

This is a question I'm struggling a couple of days with: Let $G_1,G_2$ be abelian groups, and let $H$ be a subgroup of $G:=G_1\oplus G_2$. Under what conditions must $H$ be a group of the form ...
3
votes
1answer
233 views

Decomposing the tensor product representation of $S_3$ in terms of irreducibles

I have a theorem which says that: If $\rho_1,...\rho_n$ are a complete set of irreducible $K$-representations of $S_n$ then we have that: $V^{\otimes n}=\bigoplus_1^k(V^{\otimes n}_{\rho_i})$ as ...
4
votes
0answers
83 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
1
vote
2answers
90 views

Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$

Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$ This is my attempt so ...
0
votes
1answer
89 views

Order of a permutation using its cycle decomposition

If $A=\{1,2,...,n\}$, $\Omega _A$ is the set of all permutations over $A$, $S_n=(\Omega _A, \circ)$, then for any $\sigma \in \Omega _A$, the order $m$ of $\sigma$ (Smallest $m \in \mathbb{N}$ for ...
0
votes
1answer
32 views

Is $G=\langle g,h:g^{2p}=h^2=1,g^h=g^{-1}\rangle$ for $|G|=\text{ord }g\cdot\text{ord }h$ a valid group presentation?

Let $G$ be a finite group of order $4p$ where $p\ge 5$ is any odd prime number and $\sigma,\tau\in G$ with $\text{ord }\sigma =2p$, $\text{ord }\tau=2$ and $\sigma^\tau :=\tau\sigma \tau ...
5
votes
1answer
90 views

Prove a result on the size of the minimal set that generates a finite abelian group

I am asked to prove the following: Let $G$ be a non-trivial, finite abelian group. Let $s$ be the smallest positive integer such that $G = \langle a_1,...,a_s\rangle$ for some $a_1,...,a_s \in G$. ...
2
votes
2answers
467 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
1
vote
2answers
101 views

When is $f(x)=x^2$ an automorphism of a finite group G?

I tried a few examples and found that it is an automorphism of $A_3$ Also, to satisfy the homomorphism property, $f(x)f(y)=f(xy)$, it must be true that $x^2y^2=(xy)^2$. This is true in abelian ...
8
votes
1answer
144 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
4
votes
2answers
116 views

How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?

An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then $$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$ This assertion was made in a way (i.e. ...
2
votes
2answers
137 views

The average value of irreducible character of a non-trivial finite group

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$ I try. But I think that I am wrong. ...
0
votes
1answer
57 views

Directly Computable Homomorphism

We call a homomorphism $f$ defined on a permutation or a matrix group $G$ directly computable if there is an efficient method of evaluating $f(g)$ directly from $g$, for all $g$ in $G$. I can not ...
2
votes
0answers
91 views

Groups of order 8

Ok so I am looking at a proof to all the groups of order 8, I've attached an image which basically is the start of the proof. In particular the part highlighted with yellow is causing me problem, I ...
1
vote
1answer
63 views

existence of $a \neq e$ in group such that $a^2=e$ [duplicate]

$G$ - group, $|G|=2k$, $k\in \Bbb N$ Does there exist $a\in G; a \neq e: a^2 =e$? I think I should somehow use the fact, that there is odd number of elements of $G$ which are not $e$.
3
votes
1answer
80 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
1
vote
1answer
47 views

Cyclic Subgroup?

For $U(16) = \{1,3,5,7,9,11,13,15\},$ is there a simple way to find $m \in U(16)$ such that $|m| = 4$ and $|\langle m\rangle \cap \langle 3\rangle| = 2$ and $m$ is unique without listing everything ...
0
votes
1answer
111 views

Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...
1
vote
1answer
152 views

number of p-sylow subgroups(NBHM-$2014$)

Given a finite group and a prime $p$ which divides its order, Let $N(p)$ denote the number of $p-$sylow subgroups of $G$. If $G$ is a group of order $21$, what are the possible values of $N(3)$ and ...
2
votes
1answer
63 views

Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$

Page 105 says - A careful look at Figure 6.9 reveals that the cosets of $\left< \, (0,1) \,\right>$ partition $C_3 \times C_3$. How is this true? The picture shows $gH = left picture = ...
0
votes
1answer
132 views

Visualize $A_4$ and $\langle x, z\rangle$ isomorphic to the Klein 4 group

Page 136 says Following Step 1 of Definition 7.5, the top of Figure 7.23 shows $A_4$ organized by the subgroup $\langle x, z\rangle$ (which is isomorphic to the Klein $4$ group. This reorganization ...
3
votes
2answers
172 views

Group homomorphisms and which of the following statements are true (NBHM-$2014$)

Let $G$ be a finite group of order $n\ge2$. Which of the following statements are true? a. There always exists an injective homomorphism from $G$ into $S_n$. b. There always exists an ...
3
votes
1answer
126 views

How many groups exist with order $n$ (two isomorphic groups are treated as the same group)

I have tried to solve it for the case when $n=1,2,3,4$ which i found that there is only one group for $n=1$, one group for $n=2,$ one group for$ n=3 $ and two groups for $ n=4$, then my question come, ...
2
votes
0answers
66 views

Automorphisms of finite almost simple groups

Let $P$ be a finite nonabelian simple group. Let $G$ satisfy $$ P\leqslant G \leqslant {\rm Aut}(P), $$ where $P$ is identified with $\rm{Inn}(P)$. I am trying to see if $$ {\rm Aut}(G)\cong ...
5
votes
3answers
121 views

How does one enumerate $\mathrm{Aut}(G)$, or at least compute $|\mathrm{Aut}(G)|$?

I know that the automorphisms in $\mathrm{Aut}(G)$ preserve the order of elements of $G$, so if $G$ is partitioned according to $\mathrm{Ord}$ (order), the product of the cardinalities of the ...
0
votes
2answers
38 views

In the group A/B find the order of the coset (x$_1$+2x$_3$)+B

A is an abelian free group, with the base x$_1$,x$_2$,x$_3$. will be B a sub-group that created with x$_1$+x$_2$+4x$_3$,2x$_1$-x$_2$+2x$_3$. In the group A/B find the order of the coset ...
1
vote
1answer
168 views

Group exponent properties

Is it true that for any finite cyclic group $G$, it holds that $\operatorname{exp}(G)=|G|$? My first thought was yes, since if $G$ is cyclic, then we know it has an element of order |G|, and since ...
0
votes
2answers
72 views

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.
3
votes
2answers
109 views

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even.

Given $\sigma \in S_n\setminus A_n$, prove that the order of $\sigma$ is even. I feel that I have a way to prove it: Since $\sigma \notin A_n$, the sign of $\sigma$ is $-1$. This implies that ...
4
votes
1answer
145 views

If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian.

Let $G$ be a finite group which possesses an automorphism $σ$ such that $σ(g)=g$ if and only if $g=1$. If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian. This is what I got ...
0
votes
3answers
123 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all ...
4
votes
2answers
84 views

Existence of nontrivial unit in $\mathbb{Q}[G]$, where $G$ is finite.

Suppose $G$ is a finite group of order $|G|>1$, and $\mathbb{Q}[G]$ is the group ring. I'm curious about an example of a nontrivial invertible element, i.e., one that is not of the form $ag$, ...
2
votes
2answers
70 views

Prove that a subset $\{a \in A \mid \chi (a) = 1\}$ is an $(n - 1)$-dimensional subspace of $V$.

Let $A$ be the additive group of n-dimensional vector space $V$ over the field $\mathbb{F}_p$. Let $\chi$ - a nontrivial irreducible complex character of A. Prove that a subset $\{a \in A \mid ...
2
votes
2answers
36 views

Question about number of elements in $S_p$ and number of $p$-sylow groups.

Let $G=S_p$ where $p$ is a prime. How many elements with order $p$ in $G$, and what are they? How many $p$-sylow their is in $G$? I will be glad to see a simple solution.
1
vote
5answers
150 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
1
vote
2answers
76 views

Permutation Group-$S_{10}$

How many elements of order $30$ are there in the symmetric group $S_{10}$? I worked out and got $10500$ - using Computations and adjusting each individual cycle's position ( $2-3-5$ , $3-2-5$ etc). ...
0
votes
2answers
49 views

Isomorphism class of $\mathbf{U}(p^n)$

Note that $\mathbf{U}(k)$ is the unitary group. i.e. $\mathbf{U}(k)=\{x<k | \gcd(x,k)=1\}$ We need to find the isomorphism class of $\mathbf{U}(p^n)$ where $p$ is an odd prime. The isomorphism ...
2
votes
3answers
357 views

Product of disjoint cycle

I've found the question to find product cycle of let be $\phi$ = (2, 4, 9, 7,) (6, 4, 2, 5, 9) (1, 6) (3, 8, 6) in S9. I know how to express the permutation of S9 as product of disjoint cycle, like ...
3
votes
1answer
42 views

How to find the number of transposition

I just learning the abstract algebra now, I'm stuck to find how many transpositions can be made from $(1\ 8)(2)(3\ 6\ 4)(5\ 7)$?
1
vote
1answer
38 views

How to find how many cosets are of $H \cap K$?

I'm confuse to find how many cosets of $H \cap K$ are in the G? If $G$ is a group of order 48, then $H$ of order 8, $K$ of order 6, <= $G$.
3
votes
1answer
106 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
2
votes
0answers
37 views

It is true that $\mathrm {Im}(f^{n_{0}})=\mathrm {Im}(f^m)$ for all $m\geq n_0$ implies $\mathrm {Im}(f^{n_{0}})=\{0\}$

Let $G$ be a finite abelian group, and $f: G\longrightarrow G$ an endomorphisme of $G,$ such that $\ker(f)\neq \{0\},$ and $\mathrm {Im}(f)$ a propre subgroup of $G,$ so we have a descending chain ...
1
vote
2answers
165 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
5
votes
3answers
106 views

How many Homomorphisms can be between $Z_6$ to $Z_{18}$?

How many Homomorphisms can be between $Z_6$ to $Z_{18}$? and the most important: What is the algorithm for calculating, step by step?