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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1answer
8 views

About primary decomposition of $Z_7$

I want to find the primary decomposition of $Z_7$ under multiplication group action. So I know 7 is a prime number so it is a group of order 6. So possibilities are $Z_7\cong Z_2\oplus Z_3$ However ...
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1answer
32 views

Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
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0answers
18 views

A question about a intransitive group

Assume that the intransitive group $G$ has degree $n$ and minimal degree $n-1$. If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except one are regular. Any ...
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0answers
17 views

Describe all subgroups of $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$

Is it possible to give a general discription of all subgroups of the group $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$ of inner automorphisms of $\operatorname{GL}_2(\mathbb F_{p^n})$? ...
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1answer
22 views

Subgroups in $G$ of the form $gHg^{-1}$

Let $G$ be a group and $H$ be a subgroup of finite index. Prove that there is only a finite number of distinct subgroups in $G$ of the form $gHg^{-1}$ where $g$ belongs to $G$.
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1answer
19 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
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1answer
19 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
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1answer
25 views

Prove $f(x) = x * a$ is bijective (preferably using inverse)

I have this question that I am stuck at. Let $(G; * ; I)$ be a group and let 'a' in $G$. Let $f : G \rightarrow G$ be the function defined by $f(x) = x * a$ for all $x$ in $G$. Prove that $f$ is ...
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2answers
36 views

Is this group cyclic?

Is this group cyclic, and what would be its generator? where H = {a+b(sqrt(2)) element of R | a, b element of Z} I know that in order for a group to be cyclic if the generator equals the group, but ...
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0answers
23 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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0answers
13 views

Conjugacy classes of PSL(6,7)

I need the conjugacy class sizes of projective special linear group PSL(6,7). I couldn't find it by using GAP. Could someone find it?
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1answer
33 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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2answers
24 views

Question on order of elements in groups (subgroups)

I am a bit confused at the moment, but what can we say about the order of all elements in a finite (sub)group? Suppose we have a group $G$ such that $|G|=p^k$ for a prime $p$. Next let $H$ be a ...
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2answers
17 views

Part of simple proof of nontrivial center in p-group

I'm trying to understand the proof of a Burnside theorem (as stated in Beachy's Abstract Algebra p. 328): Let $p$ be prime number. The center of any $p$-group is nontrivial. Now, In the proof they ...
2
votes
1answer
26 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...
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0answers
8 views

Characterise all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$

The title pretty much explains everything. Is it possible to give an easy characterisation of all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$?
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1answer
30 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
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1answer
20 views

Center of a Group and automorphisms

The center of a group, G is defined as: Z = {a|ag = ga for all g ∈ G}. Let $\phi$ be an automorphism of a finite group G to G. Show that for any a ∈ Z then $\phi(a) $∈ Z. Conclude that $\phi(Z) = Z$. ...
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1answer
25 views

A finite and stable part of a group is a subgroup

How to prove that a a finite and stable part H of a group G is necessarily a subgroup ? This is equivalent to proving that every element x of H has its inverse in H too :)
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0answers
21 views

Groups where there's always a “deformation retraction” homomorphism onto any subgroup.

Let $G$ be a finite group with the property that, for every subgroup $H$, there exists a homomorphism $f: G\to H$ such that $f(h)=h$ for all $h\in H$. What possible groups can $G$ be? If $P$ is a ...
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1answer
33 views

A Lagrangian group

I have a non-Lagrangian group $G$ of order $pq^3$, $Q$ a Sylow $q$-subgroup of G and a $H$ a subgroup of $Q$ with $|H|=q^2$. It is clear that $Q \subseteq N_G(H)$. I must prove that $G$ doesn't posses ...
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0answers
26 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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0answers
31 views

Cosets of $H$ in the group $S_4$

In the group $S_4$, let $H$ be the subgroup of those permutations which leave $4$ fixed: $$H=\{(1),(12),(13),(23),(123),(132)\}.$$ List all of the left and right cosets of $H$ in $S_4$.
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1answer
30 views

how to count the number of subgroups of order n with some constraints

Let $G$ be a group of order 24. How many subgroups of $G$ are there, given that there are exactly 8 elements of order 3 in $G$? Tried: since $24=2^3\times 3$, there must be a subgroup of order ...
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0answers
15 views

Find number of triangle and quadrangles in $AG_2(3)$

$\textbf{Question-}$ Show that in $AG_2(3)$ there are - i) $72$ triangles ii) $54$ quadrangles iii) $4$ triangles in each quadrangle iv) $3$ quadrangles containing a given triangle My try- I know ...
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1answer
15 views

Let $M,N$ be isomorphic as $\mathbf{Z}[G]$-mods, are they isomorphic as $\mathbf{Z}[H]$-mods, where $H<G$

So I have recently been looking at the isomorphisms of $\mathbf{Z}[G]$-mods ($G$ finite), and noticed that a couple of my examples saw them isomorphic as $\mathbf{Z}[H]$-mods also, where $H$ is a ...
3
votes
1answer
44 views

Quaternion^2 in the symmetric group

Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is ...
3
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3answers
42 views

Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
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1answer
38 views

To show a finite group G is nilpotent

Let G be a finite group and G' denotes it's commutator.If order of G' is 2.Then show that G is Nilpotent. What I have tried:G/G' is abelian,so it is nilpotent again G' is nilpotent as it's order is ...
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0answers
20 views

Finding Semi-direct products of Z/3Z and Z/7Z

This problem originated from trying to find the group isomorphisms of groups of order 21. I already worked out that there's one subgroup of order 7 that's normal, and one subgroup of order 3. Even ...
3
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1answer
21 views

Order of groups and elements

(related to this question: Finite Group and normal Subgroup) Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ We define the set $X:= \{g\in G | g^d=1\}$ ...
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1answer
31 views

In a non abelian group of order $p{^4}$ Quotient of center by commutator is abelian

Let $G$ be a non abelian group of order $p{^4}$,$p$ is a prime.Let $N$ be a normal subgroup of $G$ with |$N$|=$p$ and $G/N$ is abelian.Then prove that $N$ is a subgroup of $Z(G)$ and $Z(G)$/$N$ is a ...
2
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1answer
61 views

Finite Group and normal Subgroup

Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ and define the set $X:= \{g\in G | g^d=1\}$. Show: if $H$ is a normal subgroup of $G$ with order d then ...
2
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3answers
67 views

Find order of group given by generators and relations

Let $G$ be the group defined by these relations on the generators $a$ and $b$: $\langle a, b; a^5, b^4, ab=ba^{-1}\rangle$. I need hints how to find order of $G$.
3
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1answer
68 views

Show that the order of $G$ cannot be divisible by $7$.

Show that the order of a proper primitive group of degree $19$ cannot be divisible by $7$. It means as following- This means that a group $G$ which acts on a set $\Omega$ of cardinality $19$ (this ...
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0answers
33 views

Can a chain subcomplex be a direct summand but it's 'origin' is not? (group homology)

Let $H\!\leq\!G$ be finite groups and $C_\ast(H)\!\leq\!C_\ast(G)$ their bar complexes (each $C_k(G)$ is a free $R$-module with basis $(G\!\setminus\!\{1\})^k$). Is it possible that $H$ is not a ...
2
votes
3answers
66 views

How many distinct subgroups of order 10 are there in a non-cyclic abelian group of order 20?

We are currently working with free abelian groups and finitely generated groups. The homework problem asks us to find the number of distinct subgroups of order 10 in a non-cyclic abelian group of ...
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2answers
92 views

Order of the group.

An abelian group $G$ is generated by $x$ and $y$ with $$O(x)=16,O(y)=24,x^2=y^3$$ What is the order of $G$? My attempt:There are $24+16-1=39$ elements generated by $x$ and $y$ separately. Also ...
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0answers
68 views

Description of an abelian group

I'm again stuck in an algebra exercise. I'm not sure if I understand the problem right. Could it be that I have to show that $\mathbb{Z}[i]/\gamma$ can be expressed by a product of finite abelian ...
2
votes
1answer
35 views

Abelian subgroup of $S_n$ such that $g(a)=b$ for all $a,b \in [1,n]$

Let $G$ be an abelian subgroup of $S_n$. Suppose that for all $a,b \in [1,n]$ there is $g$ in $G$ such that $g(a)=b$. Show that the order of $G$ is $n$. I don't think that Lagrange's ...
2
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1answer
31 views

Clarifying a proof of a corollary of Lagrange's Theorem

The Corollary: If $G$ is a prime of order $p$, then $G$ is cyclic. The Proof: Let $ x \in G$, $x \neq 1$. Thus $|\langle x\rangle| > 1$ and $|\langle x \rangle|$ divides $|G|$. Since $|G|$ is ...
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1answer
29 views

Can we calculate the order of $\hom (G,G')$ in terms of $|G| $ and $ |G'|$ , when $G,G'$ are finite abelian groups?

Let $G,G'$ be abelian groups and let $\hom (G,G')$ be the set of all homomorphisms from $G$ to $G'$. We define an operation $\ast$ on $\hom (G,G')$ as: for $f,g \in \hom(G,G') \space , (f\ast ...
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1answer
45 views

Dihedral groups [closed]

Consider the Dihedral group $G=D_{12}=\langle a,b\rangle$.Which of the following is false? A) $G$ has an element of order $3$. B) All subgroups of $G$ of order $4$ are isomorphic. C) All subgroups of ...
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1answer
56 views

Is there any formula to calculate the number of normal subgroups of $S_n$?

Is there any formula to calculate the number of normal subgroups of $S_n$? Suppose i have an answer to this question it is easy to answer how many homomorphism is there from $S_n$ to any other ...
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2answers
34 views

|ab|=lcm(|a|,|b|) in an abelian group

Assume in an abelian group $G$ that $\langle b\rangle\cap \langle a\rangle=e$, then the order of $(ab)$ is the lcm of the orders of $a$ and $b$. Essentially, $|ab|=\operatorname{lcm}(|a|,|b|)$. So ...
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1answer
54 views

An iff condition for $2$-transitive groups

$\textbf{Theorem}$ - A group $G$ acts doubly transitively on a set $X$ iff $1/|G|\sum_{g\in G}|fix(g)|^2=2$. I Have no idea how to begin. If it had been finite group and finite set, then at the ...
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0answers
36 views

Solvable group of order $p^nq^m$

Let $G$ be a semi-direct product of a $p$-group and a $q$-group where $p$ and $q$ are prime number. If $G$ does not contain a normal minimal subgroup of order $q$ what we can say about $q$-sylow of ...
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votes
2answers
36 views

Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$.

Find all kinds of homomorphisms from $(\Bbb Z_n, +)$ to $(\Bbb C^*, \times)$ and from $(\Bbb Z, +)$ to $(\Bbb C^*, \times)$. Explain why they are the complete collection. My intuition is: 1) we ...
1
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1answer
30 views

Find **Pontrjagin dual.** of $\Bbb Z_n$ & $\Bbb Z$

Let $G$ be a group. Consider the set $Hom(G,\Bbb C^*)$ of homomorphisms from $G$ to $\Bbb C^*$. Define a binary operation $+:Hom(G,\Bbb C^*)\times Hom(G,\Bbb C^*)\to Hom(G,\Bbb C^*)$ s.t ...
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1answer
25 views

Stabilizer of a doubly transitive is maximal?

Is it true that if $G$ is a group acting $2$-transitively on a set $X$ , then if $x\in X$, then $G_x$ (stabilizer) is maximal in $G$. I think it must be true as a conclusion of $2$ theorems, as ...