Tagged Questions

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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0
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1answer
55 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 ...
1
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0answers
30 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 ...
0
votes
0answers
34 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 $S_4$.
1
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1answer
32 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 ...
0
votes
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 ...
1
vote
1answer
17 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
votes
3answers
43 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& ...
1
vote
1answer
41 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 ...
0
votes
0answers
22 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
votes
1answer
23 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 $X:= \{g\in G | g^d=1\}$ ...
5
votes
1answer
33 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
votes
1answer
63 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
votes
3answers
97 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
votes
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 ...
1
vote
0answers
55 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
125 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 ...
1
vote
2answers
98 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 ...
1
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0answers
71 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
36 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
votes
1answer
34 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 ...
1
vote
1answer
31 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 ...
-1
votes
1answer
46 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 ...
2
votes
1answer
57 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 ...
0
votes
2answers
35 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 ...
1
vote
1answer
56 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 ...
1
vote
0answers
44 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 ...
2
votes
2answers
37 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
vote
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 ...
1
vote
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 ...
0
votes
2answers
50 views

Permutation matrices for symmetry group $O_h = S_4 \times C_2$

Does anyone know of a quick way to enumerate the permutation matrices for the symmetry group of the cube $O_h = S_4 \times C_2$? $O_h$ has $48$ elements; if we label the vertices of the cube $1,2,…,8$ ...
3
votes
3answers
70 views

Given $o(a)=5$, prove $C(a)=C(a^{3})$

Given $o(a)=5$, prove $C(a)=C(a^{3})$ At this point I would like a hint rather than a full solution. I know we are given $a^{5}=e$ and that we wish to prove this implies that $C(a) =\{ x \in ...
1
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0answers
42 views

inner automorphisms of non-abelian simple groups

Let $G$ is non-abelian and simple group. Let $I ={\rm Inn}(G) \cong G$, $A = {\rm Aut}(G)$ and $B = {\rm Aut}(A)$. Since $Z(A)=1$, we have $A \cong {\rm Inn}(A)$, so we can identify $A$ with the ...
1
vote
1answer
53 views

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I've come up with a proof for the following question, and I'd like to know if it's correct (as I couldn't find anything online along the lines of what I did). Question Let $p$ and $q$ be primes ...
0
votes
0answers
52 views

A question on p-groups, and order of its commutator subgroup [duplicate]

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. $\textbf{TRY- }$ If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent ...
2
votes
4answers
37 views

Generators of a product of finite abelian groups

Let $n_1,...,n_r$ be positive integers. Consider the group $$G={\bf Z}/n_1 {\bf Z} \times \cdots\times {\bf Z}/n_r {\bf Z}$$ When does a given element $(k_1,\cdots,k_r)$ generate $G$? Obviously ...
1
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0answers
76 views

Is this problem still open or solved?

Problem- Let $n\geq$ and let $T$ be the set of all permutations in $S_n$ of the form $t_k=\prod_{1\leq i\leq k/2}(i,k-i)$ for $k=2,3,4.....(n+1)$. Then find the least integer $f_n$ such that ...
1
vote
1answer
69 views

A GAP code for a class of small groups

I need a GAP code for checking the following question Let $n$ be a given positive integer. Is it true that every group $G$ of order $n$ is either solvable or satisfies the condition: "For any ...
4
votes
1answer
40 views

Why Mathieu group M11 is sharply 4-transitive?

I am studying Steiner system and Construction of Mathieu groups from automorphism of some Steiner system.Mathieu group M11 is automorhism group of S(4,5,11) Steiner system. I am not able to understand ...
1
vote
1answer
44 views

Hall's Subgroup Theorem

I am currently looking into the extension of Sylow's theorems, namely through Hall-$\pi$-subgroups and Hall's Theorem. I currently have the theorem as; Let $G$ be a finite solvable group a $\pi$ be ...
2
votes
1answer
31 views

prove that $D_8 \cong C_2 \wr C_2$ .

prove that $D_8 \cong C_2 \wr C_2$ . $\wr$ is wreath product and it is using in place of $C_2 \ltimes (C_2 \times C_2)$. here is my answer : suppose $K=C_2 \times C_2$ and $C_2 \cong \langle \sigma ...
0
votes
0answers
31 views

can we express $D_{2n}$ as semi direct product?

can we express $D_{2n}$ as semi direct product? this is my answer:yes,if $D_{2n} =<a,b|a^2=e ,b^2=e ,aba^{-1}b^{-1}>$, then $<b> \triangleleft D_{2n}$ , on the other hand we have ...
-1
votes
2answers
74 views

Suppose $G$ is a group with exactly $8$ elements of order $3$. How many subgroups of order $3$ does $G$ have?

Suppose $G$ is a group with exactly $8$ elements of order $3$. How many subgroups of order $3$ does $G$ have?
3
votes
2answers
80 views

Is a group always contained in a group that surjects onto its automorphism group?

Let $G$ be a group. I am interested in embedding $G$ in a group $\tilde G$ such that there is a surjective map $\tilde G\rightarrow\operatorname{Aut}G$ whose restriction to $G$ yields the homomorphism ...
0
votes
1answer
41 views

Subgroup of an abelian group isomorphic to a given quotient group

STATEMENT: Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroups that is isomorphic to $G/H$. QUESTION: Could someone offer a proof using dual groups. I have found one ...
1
vote
3answers
66 views

Prove that no finite abelian group is divisible.

A nontrivial abelian group $G$ is called divisible if for each $a \in G$ and each nonzero integer $k$ there exists an element $x \in G$ such that $x^k=a$. Prove that no finite abelian group is ...
1
vote
1answer
47 views

Basis of vector space invariant under group action (of symmetric group)

Suppose I have a finite-dimensional real vector space $X$ and a finite group $G$ that acts faithfully on X. The task is to find a $G$-invariant basis of $X$. This means the set of basis vectors is ...
1
vote
3answers
52 views

Automorphic group of a cyclic group

Suppose $G$ is a cyclic group of order $n.$ Then $Aut(G)\cong (\mathbb{Z}/n\mathbb{Z})^\times$. Why is this true?
1
vote
2answers
71 views

Proof that the order of any finite $p$-group is a power of $p$

What is the most concise proof that the order of any finite $p$-group is a power of $p$?
4
votes
0answers
73 views

Cayley's theorem — more than one isomorphism

I've just been learning about Cayley's theorem and a couple of things occurred to me: We know that every finite group of order $n$ is isomorphic to some subgroup of $S_n$. But perhaps there are ...