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.

learn more… | top users | synonyms

2
votes
2answers
225 views

finite group whose only automorphism is identity map

Question is to prove that : A finite group whose only automorphism is identity map must have order at most $2$. What i have tried is : As any automorphism is trivial, so would be inner ...
2
votes
1answer
712 views

Finding all homomorphisms between two groups - couple of questions

Consider $\mathbb{Z}_{15}$, and $\mathbb{Z}_{18}$. Let's say I want to find all homomorphisms $f:\mathbb{Z}_{15}\rightarrow \mathbb{Z}_{18}$. I'm not interested in the answer in particular, mostly ...
2
votes
1answer
44 views

The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
2
votes
2answers
90 views

About the construction of semidirect products

I need help with the following question: We have $G_0=H_0[N]_\rho$ a semidirect product, and $\pi:H\to H_0$ an epimorphism such that $K=\ker(\pi)\unlhd H$, and $H/K\cong H_0$. We have to construct a ...
5
votes
1answer
149 views

Action on G via Automorphism

Here is an exercise from Isaacs, Finite Group Theory, $4D.1$: Let $A$ act on $G$ via automorphism, and assume that $N \trianglelefteq G$ admits $A$ and that $N \geq C_G(N)$. Assume that ...
6
votes
1answer
124 views

An exercise about p-solvable

I'm dealing with a problem about p-solvable in Isaac's finite group theory book. Question is the following: "Let $G$ be $p-$solvable and $P \in Sy{l_p}\left( G \right)$ and $K \le G$ such that $p$ ...
3
votes
6answers
368 views

Any two groups of three elements are isomorphic - Fraleigh p. 47 4.25(b)

The answer has no details. Hence maybe the answer is supposed to be quick. But I can't see it? Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$. Then because every group has ...
0
votes
2answers
105 views

Subgroups of a group

If we have a group $G$ of order $10$, and $H$ is a subgroup of $G$ then Lagrange's theorem states each $H$ may be of order $1, 2, 5$ or $10$. Now we have the trivial subgroup $\lbrace e\rbrace$ of ...
4
votes
2answers
115 views

When does $x^{n}=y^{n}$ imply $x=y$.

Let $G$ be a finite group of order $m$. Let $n$ be relatively prime to $m$. Let $x,y\in G$ such that $x^{n}=y^{n}$ prove that $x=y$. I was able to prove this result if $G$ is abelian using the map ...
2
votes
0answers
56 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
1
vote
4answers
71 views

Order of elements within a group

If $G$ is a finite group of order (size) $n$ then, for any $g \in G$, the order (period) of $g$ is a divisor of $n$. Proof: $g$ must have finite order since $G$ is finite. If the order (period) of ...
3
votes
2answers
92 views

Conjugation and generators

Let $G$ be a finite group and $x\in G$ be of order $4$. So $o(x)=4$. Suppose that all cyclic subgroups of $G$ of order 4 are conjugate. Show that there exists an involution $g\in G$ such that ...
2
votes
2answers
991 views

automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
0
votes
1answer
232 views

Number of homomorphism from $S_3$ to $\mathbb Z_6$ [closed]

Total number of homomorphism from $S_3$ to $\mathbb Z_6$ , $S_3$ to $S_4$ and how to determine?
2
votes
1answer
137 views

Irreducible representations of group of order $pq$

There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows: Suppose $p>q$. Then by the Sylow ...
9
votes
1answer
190 views

More on numbers of homomorphisms.

This is directly related to this question, but should be easier: Suppose for finite groups $G_1$ and $G_2,$ we know that for any group $H,$ $$|\rm{Hom}(G_1, H)| = |\rm{Hom}(G_2, H)|$$ Does it follow ...
1
vote
0answers
135 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
1
vote
0answers
69 views

Each irreducible representation is a subrepresentation of induced one

I'm learning what irreducible representation is and need some examples. One of them is as follows: Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation ...
1
vote
1answer
54 views

Why $PGL(2, 9)$ is not isomorphic to $S_6$?

How can I show that $PGL(2,9)$ is not isomorphic to $S_6$? My primary idea is to compare the size of conjugacy classes of two well-chosen elements in these groups. Is there another simpler ...
0
votes
1answer
46 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
4
votes
1answer
117 views

For any $n$, there are at most two simple groups of order $n$? [duplicate]

How do you prove that for any $n$ there are at most two simple groups of order $n$?
33
votes
3answers
553 views

If $\lvert\operatorname{Hom}(H,G_1)\rvert = \lvert\operatorname{Hom}(H,G_2)\rvert$ for any $H$ then $G_1 \cong G_2$

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, $\lvert\operatorname{Hom}(H,G_1)\rvert = \lvert\operatorname{Hom}(H,G_2)\rvert$. How can I show that $G_1 \cong G_2$ ?
7
votes
1answer
83 views

Proof of $|C_{G/N}(gN)| \leq |C_G(g)|$ without character theory

In the book Character Theory of Finite Groups by Isaacs the following is proven (Corollary 2.24) using character theory: Proposition: Let $G$ be a finite group and $N \trianglelefteq G$. Then ...
4
votes
2answers
123 views

Generalizing central automorphism group condition to endomorphisms

Given a group $G$, we can define its central automorphism group by $$\operatorname{Aut}_c(G)= C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)) = \{ \phi\in\operatorname{Aut}(G) : \phi(g)g^{-1} \in ...
1
vote
3answers
123 views

Subgroup of a symmetric group $S_7$

Is there any quick way to determine if $S_7$ contains a subgroup of order $6$ or $S_{11}$ a subgroup of order $30$ ? A problem carrying only 2 marks involves this. So I assume either there is some ...
1
vote
3answers
106 views

Extend a function to group homomorphism

This is maybe a trivial question. Set up Assume $S = \lbrace g_{1}, \dots, g_{n} \rbrace$ generate a group $G$ and $H$ is a finite group with elements $\lbrace h_{1}, \dots, h_{n} \rbrace$. In ...
1
vote
3answers
57 views

classification of $2$-dimensional field extensions

Let $F$ be a field and $K:F$ be a field extension such that $[K:F]=2$. Then (i). If $Char(F)\neq 2$, then there exists $\alpha\in K^*$, $\alpha\notin F^*$, such that $K=F(\alpha)$ and $\alpha^2\in ...
1
vote
0answers
60 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
2
votes
4answers
227 views

example of Frobenius endomorphisms that is not automorphism

Let $F$ be a field of characteristic $p$ and $F$ has infinitely many elements. Prove that the map $\sigma: \alpha\mapsto \alpha^p$ is a field endomorphism but not necessarily automorphism. How to ...
3
votes
1answer
117 views

Determining All Groups of Order 308

I just turned in an exam today and I wanted to answer this question, but I couldn't so I had to choose another (you could omit one question). Up to isomorphism, we had to determine all groups of ...
3
votes
3answers
159 views

On the 11 Sylow subgroup of a group of order 792

It seems that the 11-Sylow subgroup of a group of order $792=11×2^3×3^2$ is normal. Could you help me why it is true
-2
votes
1answer
290 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
3
votes
2answers
66 views

minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
1
vote
0answers
42 views

Irreducible representations and abeliean subgroups

There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of abelian subgroup ...
5
votes
0answers
151 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
1
vote
1answer
65 views

Let $\tau_1, \tau_2 \in A_6$ be permutations with cycle-type $(2, 4)$. Show there exists $\tau \in A_6: \tau \tau_1 \tau^{-1} = \tau_2.$

Let $\tau_1, \tau_2 \in A_6$ be permutations with cycle-type $(2, 4)$. Show there exists $$\tau \in A_6\mid \tau \tau_1 \tau^{-1} = \tau_2$$ I know $\tau_1 = (a b c d)(e f)$ and $\tau_2 = ...
2
votes
0answers
68 views

Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$.

Let $G$ be a finite group, $N$ be a normal subgroup of $G$. Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$. ...
0
votes
3answers
111 views

Is this identity for the Dihedral group correct?

Let $D_n$ represents the Dihedral group with $2n$ elements, and my question(based on some physics backgrounds) is: Does $Z_2$ a normal subgroup of $Q_8$? If it is, then is the indentity $D_2\cong ...
20
votes
6answers
538 views

Generic elementary group theory problems.

This question is about generic group theory problems. here are examples for what I’m referring to: Prove that any group of order $p^2$, where $p$ is a prime, is abelian. Let $G$ be a ...
0
votes
0answers
70 views

Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...
0
votes
1answer
27 views

Diffie helman on additive group

Given the addivite group G, with |G| = n and generator = g, what are the computations and exchanged messages for Diffie-Hellman? I'm not sure because the group is additive.
1
vote
1answer
84 views

Prove that, if all maximal subgroups of a finite group are abelian, at least one of maximal subgroups is normal

Prove that if all maximal subgroups of a finite group are abelian, at least one of maximal subgroups is normal Help me some hints
-1
votes
2answers
138 views

Group acting on a set.

Let $G$ be a group of order $7$ acting on a set of $5$ elements. Show that the action of $G$ must have a fixed point.
3
votes
1answer
80 views

If $G$ is a nonabelian finite simple group, does $G$ contain a maximal subgroup which is not abelian?

Follow this link a nonabelian group whose every proper subgroup is abelian is not simple. So if $G$ is a nonabelian finite simple group, does $G$ contain a maximal subgroup which is not abelian?
2
votes
2answers
215 views

Sylow p-subgroup of a direct product is product of Sylow p-subgroups of factors

Let $G$=$HK$ be a finite group (direct product), $P$ a Sylow $p$-subgroup of $G$. Prove that $P$ = $H'K'$, where $H'$ and $K'$ are Sylow $p$-subgroups of $H$ and $K$ respectively. I am very new ...
8
votes
1answer
105 views

Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
7
votes
0answers
202 views

The order of $H$ is relatively prime to its index $[G:H]$

Suppose that a subgroup $H$ of a finite group $G$ satisfies one of the following two conditions: (i) For any nonidentity element $x$ of $H$ we have $C_{G}(x) \subset H$ (ii) If $K$ is a subgroup of ...
2
votes
1answer
175 views

How to determine the Galois group of irreducible polynomials of degree $3,4,5$

Let $f$ be an irreducible (over $\mathbb{Q}$) polynomial in $\mathbb{Z}[x]$, $\deg (f)=3,4,5$. The Galois group of an irreducible polynomial $f\in \mathbb{Z}[x]$ acts transitively on distinct roots in ...
3
votes
3answers
120 views

A well-known theorem of O. Schmidt

Prove that if all the proper subgroups of a finite group $G$ are nilpotent, then $G$ is soluble. How to I prove it? Thanks in advance.
0
votes
1answer
95 views

Maschke's theorem and the problem of the irreducible representation

Need to prove the following statement Let $\rho_k:<a>_n\rightarrow GL_2(R)$ is representation. $\rho_k(a)= \left( \begin{array}{cc} \cos {\frac{2 \pi k}{n}} & -\sin{\frac{2 \pi k}{n}} \\ ...