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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5
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1answer
282 views

Image of conjugacy class under surjective homomorphism

There is a surjective homomorphism from $G$ to $G'$. Let $C$ denote the conjugacy class of element $x$ in $G$, $C'$ the conjugacy class of the image of $x$ in $G'$. Prove the order of $C'$ divides the ...
8
votes
3answers
145 views

Subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$

what are the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$? I know that there are three such subgroups, and two subgroups are clear to me, namely the subgroup isomorphic to ...
0
votes
1answer
319 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
5
votes
3answers
265 views

$G$ group, $H \trianglelefteq G$, $\vert H \vert$ prime, then $H \leq Z(G)$

Let $G$ be a finite group. Let $H \trianglelefteq G$, with $\vert H \vert = p$, a prime, where $p$ is the smallest prime dividing $\vert G \vert$. Prove that $H \leq Z(G)$. (Hint: If $a \in H$, by ...
3
votes
1answer
131 views

The cancellation property for finite abelian groups

I need some hints to prove that: Let $A,B,C$ are finite abelian groups such that $A\oplus B\cong A\oplus C$. Prove that $B\cong C$. I know that every finite abelian group can be written as a ...
2
votes
2answers
99 views

Coercion in MAGMA

In MAGMA, if you are dealing with an element $x\in H$ for some group $H$, and you know that $H<G$ for some group $G$, is there an easy way to coerce $x$ into $G$ (e.g. if $H=\text{Alt}(n)$ and ...
2
votes
2answers
289 views

Finding generators of an automorphism group?

I'm trying to construct a homomorphism $\theta:C_2 \rightarrow AutC_{17}$. To do this I need to map the generator of $C_2$ (call it $a$) to a generator of $AutC_{17}$, but to do this I need a way of ...
4
votes
1answer
203 views

Examples: Representations over finite rings and Maschke's theorem

Is there a possibility to get the simple $R[G]$-modules, if $R$ is the ring $\mathbb{Z}/n\mathbb{Z}$, $G$ a finite group and $\operatorname{ord}(G)$ and $n$ are relatively prime? For which groups ...
7
votes
2answers
203 views

Why is the minimum size of a generating set for a finite group at most $\log_2 n$?

It seems to be known that the minimum size of a generating set for a finite group of order $n$ is at most $\log_2 n$. Can someone explain why this is true? Edit: noted that the logarithm is base 2, ...
0
votes
1answer
59 views

Commutator group of parabolic subgroups of $GL_n(q)$

Let $q$ be a prime power, $G=GL_n(q)$ and $P=q^{km}{:}(GL_k(q) \times GL_m(q))$ be a parabolic subgroup of $G$, where $k+m=n$. What is the commutator group $P'$ of $P$?
0
votes
3answers
627 views

Every group has a subgroup of prime order?

Is there a quick proof that given any finite group $ G $ with $ |G| = n$, it has a subgroup of prime order $ p \geq 2$? I've managed to prove the statement by writing down the unique prime ...
0
votes
0answers
113 views

A reference writen in Russian

I'm reading a paper "Groups that can be represented as a product of two solvable subgroups" published in 1986 in Comm. Algebra. Since I do not understand Russian, I only read the abstract in this ...
6
votes
3answers
780 views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
4
votes
1answer
947 views

Existence of normal subgroups for a group of order $36$

Prove that a group of order 36 must have a normal subgroup of order 3 or 9. Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then n must be 1 or 3. Let n3 be the number of 3-Sylow ...
1
vote
2answers
352 views

kernel of a homomorphism

Let $G$ be a group. Then $G$ acts on itself by conjugation, which corresponds to a homomorphism $K\colon G\to\operatorname{Aut}(G)$. Show that the kernel of $K$ is $Z(G)$. $K: G\to ...
1
vote
0answers
47 views

What are various approaches of proving that a permutation group is a subgroup of another?

I have two permutation groups $G_1=\langle g_1,g_2\rangle$ and $G_2=\langle h_1,h_2\rangle$ acting on $\Gamma_1$ and $\Gamma_2$ respectively. I want to prove that $G_1 \leq G_2$. What are more ...
1
vote
1answer
78 views

Seeking a proof of: If any two Abelian groups of order $d$ are isomorphic, then $d$ is squarefree.

Suppose that $d \in \mathbb{N}$ satisfies the property that given any two Abelian groups $A_1, A_2$ of order $d$, $A_1 \cong A_2$. Prove that given any prime $p$, $p^2 \nmid d$. What can one say for ...
0
votes
1answer
74 views

showing G is abelian

If $G$ group of order $52$ includes a normal group of order $4$ then $G$ is abelian. I did like this $$ |G|=52=2^2\cdot 13 $$ let $H$ be normal group of order $4$. $n_{13}=1$ thus $G$ has a $K$ ...
0
votes
3answers
90 views

order of elements in a finite group

If $|G|=p^rm$ with $(p,m)=1$, suppose that $x\in G$ is an element such that $o(x)=p^{r_1}m_1$ with $r_1>0$ and $(m_1,p)=1$. I dont understand why exist $a,b\in G$ such that: 1) $a$ has order a ...
0
votes
0answers
119 views

Complement of a subgroup

Let $G$ be a finite group. Suppose that every element of order $2$ of $G$ has a complement in $G$, then $G$ has no element of order $4$. Proof. Let $x$ be an element of $G$ of order $4$. By ...
4
votes
2answers
275 views

Where can I find the original papers by Frobenius concerning solutions to $x^n = 1$ in a finite group?

A theorem proven by Frobenius states that If $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Articles discussing this theorem ...
3
votes
2answers
255 views

Adapting a proof on elements of order 2: from finite groups to infinite groups

Consider the following problem, appropriate for a first course in Group Theory: Problem: Prove that there cannot be a group with exactly two elements of order $2$. General Proof: Suppose for the ...
2
votes
0answers
58 views

A bound for the exponent of the Schur multiplier of group G

Let $G$ be a finite group and $M(G)$ be the Schur multiplier of $G$. Show that the exponent $\exp(G)$ of $G$ divides the product of the exponents $\exp(M(S_p))$ of the Sylow $p$-subgroups $S_p$ of ...
0
votes
1answer
97 views

If $a^{3^3}=b^9=1$ for generators $a,b$ of $G$, can we conclude that $G$ is a $3$-group?

I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$. Are these informations sufficient to affirm that the group is a $3$-group? Adding the relation $b^{-1}ab=a^4$, can we ...
5
votes
3answers
399 views

Subgroup generated by Sylow p-subgroups is normal.

This is one part of a homework question. If we show this fact, then the rest of the problem is solved. Let $G$ be a finite group and let $H$ be the subgroup generated by all Sylow p-subgroups. We ...
141
votes
2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
0
votes
1answer
51 views

Prove K is a normal subgroup of An for some integer n

I am given a set K with some given values and want to show that it is a normal subgroup of An for some given integer n. Is this how i prove it? First prove K is a subgroup of An Second prove that An/K ...
1
vote
1answer
76 views

existence of a normal subgroup

let the G group of order 12. show that G has normal subgroup of order 3 or 4. by showing G is not basic group, G has normal groups besides e and G. I know it is simple but I dont know what to do ...
3
votes
2answers
279 views

Indecomposable modules

Suppose $q$ is a prime $(\neq 2)$ and $G$ a finite group, for example the cyclic group $C_p$. Is there a way to determine all the $\textbf{indecomposable}$ $\mathbb{F}_{q^n}[G]$ modules for some $n\in ...
22
votes
1answer
660 views

Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How ...
3
votes
1answer
66 views

Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.

As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the ...
2
votes
2answers
259 views

Homomorphisms from $S_4$ to $Z_2$

Suppose $\phi : S_4 \rightarrow Z_2$ is a surjective homomorphism. Find $\ker\phi$. Determine all homomorphisms from $S_4$ to $Z_2$. My solution: since $\phi$ is surjective, then by the first ...
3
votes
1answer
217 views

A conjugacy class $C$ is rational iff $c^n\in C$ whenever $c\in C$ and $n$ is coprime to $|c|$.

Let $C$ be a conjugacy class of the finite group $G$. Say that $C$ is rational if for each character $\chi: G \rightarrow \mathbb C$ of $G$, for each $c\in C$, we have $\chi(c) \in \mathbb Q$. I am ...
3
votes
0answers
101 views

Commutators in finite groups

I get stuck with this question : Let $g$ be an element in a finite group $G$, and let $k$ be an integer coprime to $|g|$. Prove that $g$ is a commutator in $G$ if and only if $g^k$ is a commutator. ...
3
votes
2answers
344 views

$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$

If $H \trianglelefteq G$, need $G$ contain a subgroup isomorphic to $G/H$? I worked out the isomorphism types of the quotient groups of $S_3, D_8, Q_8$. For $S_3$: $S_3/\{1\} \cong S_3$, ...
0
votes
1answer
56 views

Does there exist a perfect group with a subgroup of index at most 4?

Does there exist a finite perfect group with a subgroup of index at most 4?
4
votes
1answer
88 views

The extension of $PSL_2(q)$ by its outer automorphism group

Let $q=p^f$ be a prime power. Is $P\Gamma L_2(q)$, the automorphism group of $PSL_2(q)$, a semidirect product of $PSL_2(q)$ by its outer automorphism group $Z_{\gcd(2,q-1)}\times Z_f$? If it is not in ...
2
votes
3answers
101 views

If $\#A$ and $m$ are relatively prime, then $a\mapsto ma$ is automorphism?

Is it true if $A$ is a finite, abelian group and $m$ is some integer relatively prime to the order of $A$, then the map $a\mapsto ma$ is an automorphism? It's left as an exercise in some course ...
1
vote
1answer
210 views

Do finite $p$-groups have subgroups of all possible orders containing a given non-normal subgroup?

It is well known that finite $p$-groups have (normal) subgroups of all possible orders. Now, what can we say about subgroups containing a given non-normal subgroup? i.e. Let $G$ be a group of order ...
1
vote
3answers
74 views

Product of two elements of order q

Let $G$ be finite group. Let $x$ and $y$ be two elements of order a power of $q$, where $q$ is prime. Is the order of $xy$ equal to a power of $q$ (or of order 1)? thanks!
15
votes
2answers
491 views

Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?

A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
0
votes
0answers
91 views

finite p-groups admit a central series

If a central series is considered as $$G = G_0 \supset G_1 \supset \cdots \supset G_m = \{1\}$$ such that $$G_{i+1} \triangleleft G_i$$ and $$G_i/G_{i+1} \subset Z(G/G_{i+1})$$ then, Show that finite ...
1
vote
2answers
444 views

Every Simple Abelian group is cyclic of prime order?

This was in a claim in my class notes in a proof that every Solvable Simple group is of prime order. I was able to verify it in the case where $G$ is finite, which I think might be a missing ...
11
votes
0answers
345 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
0
votes
0answers
14 views

How small can a $\mathcal{C}_1$ subgroup of $PSL_2(q)$ containing elements of certain prime orders be?

Let $q=p^f$, $r$ be a primitive prime divisor of $p^f-1$, i.e., $r\mid p^f-1$ but $r\nmid p^j-1$ for $j<f$. Let $G=Z_p^f:Z_\frac{q-1}{\gcd(2,q-1)}$ be the parabolic subgroup of $PSL_2(q)$, i.e., ...
13
votes
2answers
749 views

Is finite group theory still a fruitful area of research?

A while back I was learning about finite groups in an algebra class. I mentioned to a friend that finite group theory might be an interesting area of research to pursue. She asked something along the ...
5
votes
1answer
235 views

Normal Subgroups in a p-group

How can one prove the following claim: Elementary abelian $p$- group of order $p^n$ have the maximal number of normal subgroups among all $p$-groups of the same order. Is is indeed true? ...
5
votes
1answer
57 views

Something behind the substitution $h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^2_{t}$?

I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups. In the first and second chapter, one trick he uses quite often is the substitution ...
1
vote
4answers
2k views

Any subgroup of index $p$ in a $p$-group is normal.

Let $p$ be a prime number and $G$ a finite group where $|G|=p^n$, $n \in \mathbb{Z_+}$. Show that any subgroup of index $p$ in it is normal in $G$. Conclude that any group of order $p^2$ have a normal ...
2
votes
1answer
252 views

p-group: cyclic $n \leq 1$ | abelian $n \leq 2$

$p$ prime number, $n$ a non-negative integer, $G$ group. (a) $\forall G$ with $|G| = p^{n}$ cyclic $\Leftrightarrow$ $n \leq 1$. (b) $\forall G$ with $|G| = p^{n}$ abelian $\Leftrightarrow$ $n \leq ...