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
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2answers
144 views

Greatest elements in crystallographic root systems

I have a question regarding a remark in the book "Reflection Groups And Coxeter Groups" by James E. Humphreys (unfortunately the book is not to be found as a whole on google books or such). In ...
5
votes
3answers
162 views

$G'$ and the product of all the elements in an odd order group

Yesterday, through working on a question Groups with only one element of order 2, Don antonio brought out a nice question within the comments: The product of all the elements in an odd order group ...
2
votes
2answers
252 views

Isomorphic to the Klein 4 group

I need to prove the group of the units of $\mathbb{Z}_3\times\mathbb{Z}_3$ is isomorphic to the Klein-4 group. But I'm really struggling to prove this. Any hints to start me off in the right ...
0
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1answer
76 views

G-Invariant; Group Cohomology question

I think this is a pretty simple question, but I cannot seem to find the answer anywhere: In the context of group cohomology of a group $G$, $G$-invariant maps are often mentioned. What is the ...
17
votes
3answers
567 views

How “abelian” can a non-abelian group be?

Something I have been wondering: in general, is there a bound for how many elements in a finite non-abelian group $G$ can commute with every other element? Equivalently, is there is a bound for the ...
1
vote
1answer
67 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
0
votes
1answer
66 views

simplicity of G

I took an exam today. If I remember correctly question was like this: let G be a group. if it has "a" element which has exact two conjugates, then G cant be simple. I answered: let that two ...
1
vote
1answer
108 views

Example of infinite group with infinitely many simple subgroups

What is an example of an infinite group with a composition series and infinitely many simple subgroups?
1
vote
1answer
81 views

Set of Sylow subgroups of a finite group

Let $G$ be a finite group, $p$ a prime dividing the order of $G$, and let $S=\{ P: P \text{ is a Sylow }p \text{-subgroup of }G\}$. Let $x\in G.$ Then $S=S^{x}.$ Proof. Since the elements of $S^{x}$ ...
1
vote
1answer
30 views

Using normal series of a group to build it up via extensions

In the context of normal and subnormal series I've found the following: "From a finite subnormal series of a group $G$ we obtain a sequence of exact sequences and thus $G$ is built up out of the ...
1
vote
2answers
120 views

Generators of a finite additive cyclic group

Let $C$ be an abelian additive group and write e for a generator of $C$. The elements of $C$ are then $0,e,2e,3e,\dots,(n-1)e$. If $C$ is finite, prove that the element $ke$ is another generator of ...
0
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1answer
222 views

Quotient groups of a finite symmetric group

What are the quotient groups of a finite symmetric group $S_n$? Can we classify them?
0
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1answer
121 views

Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.

Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points. Using Fermat's ...
4
votes
2answers
199 views

What does Frattini length measure?

I have heard derived length, for example, described as a measure of "how non-commutative" the group is. An abelian group will have derived length $1$, whereas a non-solvable group will be so ...
2
votes
1answer
292 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
4
votes
1answer
1k views

If $H$ is a proper subgroup of a $p$-group $G$, then $H$ is proper in $N_G(H)$.

Let $H$ be a proper subgroup of $p$-group $G$. Show that the normalizer of $H$ in $G$, denoted $N_G(H)$, is strictly larger than $H$, and that $H$ is contained in a normal subgroup of index $p$. ...
2
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0answers
194 views

Normal Sylow subgroups in a group of square free order

If $|G|=n$ with $n$ square free then there exists at least a normal Sylow subgroup? Any suggestion are welcome. Thanks.
4
votes
1answer
322 views

If a finite group has $\geq p+1$ Sylow $p$-subgroups (of order $p^n$), then there are $\geq p^{n+1}$ elements in the Sylow $p$-subgroups

Suppose $G$ is a finite group with Sylow $p$-subgroups of order $p^n$. I am trying to show that if there are $\geq p+1$ Sylow $p$-subgroups, then the union of all these subgroups has order $\geq ...
3
votes
1answer
220 views

Permutation representations of general linear groups over finite fields

Let $q=p^f$ be a prime power, $V$ is a $n$-dimensional vector space over $GF(q)$ and $G=GL(n,q)=GL(V)$. Is every transitive permutation representation $\rho$ of $G$ on $q^n-1$ points isomorphic to the ...
6
votes
1answer
153 views

Number of solutions of $x^d = 1_G$ .

Let $G$ be a finite group (not necessarily abelian). Let $d \mid |G|$, where $d \in \mathbb{N}$. Does there exist at least $d$ solutions to the equation $x^d = 1_G$ for $x \in G$? I can prove it for ...
5
votes
3answers
232 views

Automorphisms of $\mathbb{Z}_2 \times \mathbb{Z}_2$

Okay so I need to compute an automorphism on $\mathbb{Z}_2\times\mathbb{Z}_2$ using the fact that if $f\colon G\to H$ is an automorphism and $G=\langle K\rangle$, then $f$ is determined by where it ...
2
votes
2answers
527 views

Prove that there are no simple groups of order 224.

Prove that there are no simple groups of order 224. Let $G$ be a finite group such that $\vert G \vert = 224 = 2^5 \cdot 7$. We know that $n_2 \mid 7$ and $n_2 \equiv 1 \pmod 2$ and we know that $n_7 ...
4
votes
1answer
175 views

Constructing Semi-Direct Products

I need to decide how many semi-direct products $H\rtimes Q$ can be constructed for $H=C_{42}, Q=C_{3}$ where $C_{n}$ denotes the cyclic group of order n. I know I need to be finding homomorphisms ...
2
votes
1answer
126 views

$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$

Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
5
votes
1answer
254 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
139 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
264 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
263 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
118 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
88 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
231 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
188 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
172 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
57 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
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3answers
558 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
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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
588 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 ...
5
votes
1answer
666 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 ...
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vote
2answers
308 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 ...
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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
72 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
87 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 ...
2
votes
2answers
238 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
238 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 ...
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0answers
57 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 ...
4
votes
3answers
337 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 ...
136
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 $G\rightarrow H$?