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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173 views

Group theory - subgroups

Let $G=(\mathbb{Z}_n,+)\,$ be a group where $\,n \geq 2$ and $d \neq 0$. Find all possible values for $d$ and $n$ so that $\{0,d\}$ is a subgroup of $G$. I know that for subset of $G$ to be a group ...
4
votes
1answer
288 views

The inverse of Lagrange's Theorem is true for finite supersolvable group.

We have already known that the inverse of Lagrange's Theorem is a right fact about for example abelian or nilpotent finite groups. How can I show that: If $G$ be finite and supersolvable$^*$ and ...
86
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1answer
2k views

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
4
votes
3answers
2k views

S4/V4 isomorphic to S3 - Understanding Attached Tables

I think I see $ S_4/V_4 \cong S_3 $ from the first table beneath marked in the green. I just ignore $ V_4 $ and think of it as mapped away by the bijection $ f^{-1} $ where $ f(s) = s V_4 \iff ...
3
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0answers
142 views

Is there a simple way to show that wreath product is associative?

Is there a simple way to show that wreath product is associative? If your proof is short, please write it explicitly. Thank you.
3
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2answers
5k views

Proof of Lagrange theorem - Order of a subgroup divides order of the group

The Lagrange theorem states: If $G$ is a finite group, and $H$ a subgroup of $G$, then the order of $H$ will divide the order of $G$. More precisely, $|G| = |H| \cdot (\mathrm{number\, of\, ...
1
vote
1answer
225 views

Existence and structure of a group of order $p^2q$ where $p\mid q-1$ from a given presentation.

Let $p$ and $q$ be integer primes such that $p$ divides $q-1$. (a) Show that there exists a group $G$ of order $p^{2}q$ with generators $x$ and $y$ such that $x^{p^{2}} =1$, $y^{q}=1$, and ...
4
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1answer
64 views

On the structure of maximal parobolic subgroups of orthogonal groups over finite fields

Let $q=p^f$ be an odd prime power and $P$ be a maximal parobolic subgroup of $GO^\varepsilon(n,q)$ stabilising a totally singular $k$-subspace. It is known that $P$ has shape $A{:}(B\times C)$, where ...
4
votes
2answers
570 views

Number of elements of given order in a group

Example 1: "Calculate the number of elements of order 2 in the group $C_{20} \times C_{30}$" To do this, I split the groups into their primary decompositions and got that the groups with elements of ...
1
vote
1answer
208 views

Calculating the number of elements of some order of a direct product of groups

My question is: Consider a finite group $G$. For any integer $m \geq 1$ set $\gamma(m) = \gamma_G(m)$ the number of elements $g \in G$ such that ord($g$) = $m$. We say that $m$ is a possible order ...
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5answers
385 views

In which of the finite groups, the inverse of Lagrange's Theorem is not correct?

This is a multiple choice for finite groups. For which one of the following groups, the converse of Lagrange's Theorem is not generally satisfied? I know the converse is true for cyclic groups. 1) ...
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vote
1answer
160 views

An example of a group of order 336, not isomorphic to $PGL(2,7)$.

I need an example of a finite group $G$ by the following properties: 1) Order $G$ is $336$. 2) For every prime $p$, $G$ has not any elements of $7p$. 3) the number of Sylow $7$-subgroups $G$ is ...
1
vote
2answers
241 views

Group of order $24$ containing no elements of order $6$

Let $G$ be a group of order $24$ with no elements of order $6$. Let $T$ be a subgroup of $G$, is a Sylow $3$-subgroup. I have prove that $G$ has no normal subgroup of order 2, so it is also clear that ...
7
votes
2answers
174 views

Very generic question about Commutator and Center

Let $G$ be a finite group and $G'$ the commutator group of $G$. What can I say about $G' \cap Z(G)$? Could you be as specific as possible about p-Groups?
1
vote
1answer
250 views

Structure of the centralizer of an element in Sym(n)

Do we know the structure of the centralizer of any element in $S_n$? Or the centralizer of a permutation which has $q$ disjoint $r$-cycles where $r\leq n$. If we know, can anyone give proof or ...
4
votes
3answers
310 views

group multiplication table

I really looked all over the web and searched for an example I will understand. I don't understand how to complete a multiplication table! (all examples I found the Identity element was given) ...
5
votes
5answers
178 views

Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?

Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$. I need to check if this is a group and if it does than is it abelian or not and finite or not. Well... first, I'm not sure if this is a group. for ...
1
vote
1answer
64 views

Let $N$ be a normal subgroup of group $G$ and $G=(N\times C_{3})\rtimes C_{2}$. Then prove $G=N\times (C_{3}\rtimes C_{2})$.

Let $N$ be normal subgroup of $G$ and $G=(N\times C_{3})\rtimes C_{2}$. Then prove $G=N\times (C_{3}\rtimes C_{2})$. Thank you
8
votes
2answers
375 views

Nonabelian $p$-groups all of whose proper subgroups are abelian.

Theorem. Let $G$ be a finite, non-abelian $p$-group all of whose proper subgroups are abelian. Then $|G'|=p$. Take a counterexample of minimal order. Assume that exist a $H$ such that ...
2
votes
2answers
124 views

1 dimensional representations of $S_n$

I want to show that $S_n$ has only two 1 dimensional represnetations. mainly the trivial and sign represnetations. Where I assumed that our Field we're working on is with characteristic $\neq 2$. ...
4
votes
0answers
96 views

Prime divisor in the Automorphism group

Suppose that $G$ is a non-abelian simple group and $r$ is a prime number such that $r$ not divide $|G|$. It is a question for me that whether $r$ can divide |Aut($G$)|?
7
votes
1answer
604 views

When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a ...
5
votes
2answers
160 views

Subgroup of a soluble group is soluble

I'm trying to show that if $G$ is a soluble group with $H$ some subgroup then $H$ is also soluble. My argument is as follows: As $G$ is soluble then we have the subnormal series: $\{e\}\triangleleft ...
3
votes
1answer
168 views

When are $((C_2 \times C_2) \rtimes C_3) \rtimes C_2$ and $((C_2 \times C_2) \rtimes C_2) \rtimes C_3$ isomorphic?

Let's consider $\mathfrak{G}:=((C_2 \times C_2) \rtimes_{\phi} C_3) \rtimes_{\nu} C_2$ (which I do believe is $\mathcal{S}_4$, please confirm or argue against) and $G:=((C_2 \times C_2) \rtimes_{\mu} ...
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votes
3answers
2k views

Constructions of the smallest nonabelian group of odd order

I write $|X|$ for the number of elements in a finite set $X$. Recall some basic facts: If $p$ is a prime number, then any group $G$ of order $p^2$ is abelian. Sketch of proof: Fix a prime $p$ ...
7
votes
1answer
135 views

Bound on the number of p-groups for fixed exponent

It's well-known that for each prime number $p$ there are exactly two groups of order $p^2$, five of order $p^3$, and fifteen of order $p^4$ (at least when $p>3$). I know that the classification of ...
4
votes
1answer
369 views

Showing that a cyclic group of prime power order has only 1 composition series

I am trying to show that a cyclic group of prime power order has only 1 composition series. Is the following correct? Let $G=C_{p^n}$. Then as cyclic groups are abelian we have that there is a ...
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1answer
367 views

A question about intersection of center and commutator subgroup

Let $G$ be a finite group such that $G'\cap Z(G)\neq 1$. Suppose also that $G'$ is an elementary abelian $p$-group; $G'\nleq Z(G) $; $(G/Z(G))'$ is a minimal normal subgroup of $G/Z(G)$. Can we ...
5
votes
3answers
447 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
1
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1answer
64 views

A group that has a $\frac{3}{2}$-transitive subgroup

Do you know a group that has a $\frac{3}{2}$-transitive subgroup and it is not $\frac{3}{2}$-transitive itself?
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3answers
3k views

How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? [duplicate]

How do I find the number of group homomorphisms from the symmetric group $S_3$ to $\mathbb{Z}/6\mathbb{Z}$?
2
votes
1answer
65 views

a question on congruences

I would like to prove that the following 2 are equivalent: $\gcd(a,n)=1$ and $\exists x: x^m\equiv a \pmod n$ $a^{\frac{\phi(n)}{d}}\equiv 1 \pmod n$ where $d=\gcd(m,\phi(n))$ $\phi(n)$ is Euler ...
2
votes
2answers
367 views

Let $G$ be a group of order $56$. Then which of the following are true

Let $G$ be a group of order $56$. Then which of the following are true All $7$-sylow subgroups of $G$ are normal All $2$-Sylow Subgroups of $G$ are normal Either a $7$-Sylow subgroup or a ...
9
votes
1answer
635 views

Find the number of homomorphisms between cyclic groups.

In each of the following examples determine the number of homomorphisms between the given groups: $(a)$ from $\mathbb{Z}$ to $\mathbb{Z}_{10}$; $(b)$ from $\mathbb{Z}_{10}$ to ...
2
votes
1answer
633 views

Rotman introduction to theory of groups exercise

From Rotman "Introduction to the Theory of Groups", ex. 2:54: Let $ G $ be a finite group, and let $H$ be a normal subgroup with $(H,[G:H])=1$. Prove that $H$ is the unique such subgroup in G. ...
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1answer
67 views

proving that a action of hopf algebra k(G) on A implies a G-grading on A

Let $k(G)$ be the hopf algebra of functions on $G$ with values in $k$ with pointwise multiplication and a comultiplication given by $\Delta(f)(x,y) = f(xy)$ and let $A$ be a $k$-algebra. I read (in ...
3
votes
1answer
115 views

Peculiar presentation of Symmetric group of degree 10

In the context of some problem I am working on, I got this peculiar presentation of a group. I have established computationally that this group is $S_{10}$, but I was wondering if it can be done ...
0
votes
1answer
103 views

Let $P$ be a Sylow $p$-subgroup of $\operatorname{Sym}(n)$. If $p$ does not divide $n$, then $P\leq\operatorname{Sym}(n-1)$

Let $P$ be a Sylow $p$-subgroup of $\operatorname{Sym}(n)$. If $p$ does not divide $n$, then $P\leq\operatorname{Sym}(n-1)$. I can compute the order of $P$ and see that there exists a Sylow ...
2
votes
2answers
303 views

Why is the order of an element in the group of units mod n , $U(n)$, equal to the totient of n?

I came across this statement in an abstract algebra textbook, I am looking for a proof. EDIT: I guess I was not clear, what I meant was If $k\ \epsilon\ U(n)$ and $m\ \epsilon\ \mathbb{Z}^+$ and ...
3
votes
0answers
85 views

Equality in commutator subgroup

I will start by apologizing if this questions seems twisted. I am reading the paper Cohomology theory of groups with a single defining relation (Lyndon, $1950$) and the question comes from page $659$ ...
5
votes
0answers
106 views

$H=\langle a,b| a=bab, b=aba\rangle $ and $\frac{H}{A}\cong\ Q_8$

Here is my problem: Let $$H=\langle a,b| a=bab, b=aba\rangle $$ and $\frac{H}{A}\cong\ Q_8$ wherein $A\leq Z(H)\cap H'$. Show that $H\cong Q_8$. Working on the elements, I could see that ...
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vote
1answer
150 views

Finite simple group with subgroups of same order

Let $D$ be a finite simple group with $H < D$ and $K < D$. Also $[D:H]=q$ and $[D:K]=p$, where $p$, $q$ are primes. Want to show that $p=q$. I want to come up with a contradiction with one of ...
0
votes
1answer
78 views

Group of order three distinct primes

I want to show that a group G of order 345 is Abelian. I used Sylow's theorem to find Syl(5)=Syl(23)=1. but i was unable to conclude Syl(3)=1 because i found Syl(3)=1 or 115. I'm not sure how to ...
0
votes
2answers
538 views

How would I show that there are no onto homomorphisms from Q8 to Z4?

I have to show that there are no onto homomorphisms from _ to _, including Q8 to Z4. How would I show that? Also can someone explain what onto means again, my teacher didn't explain it well to me.
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vote
1answer
113 views

A set of non-isomorphic finite groups is a finite set

Let F= set of all non-isomorphic groups of order n where n>=2. I want to show that F is a finite set. I want to use the fact: Every group |G|=n is an isomorphic to a subgroup of Sn. But i don't know ...
2
votes
2answers
419 views

Abelian groups of order $14,27,30,$ and $21$.

Which of the following statements is false? Any abelian group of order $27$ is cyclic. Any abelian group of order $14$ is cyclic. Any abelian group of order $21$ is cyclic. Any abelian group of ...
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vote
1answer
162 views

How do you Classify all such groups?

Assume that: $G$ contains a normal subgroup $H$ of order $9$, and $G$ is generated by $H$ and an element $x\in G-H$ of order $3$. How to classify all such groups $G$? I think $9$ divides the ...
2
votes
1answer
153 views

How to describe automorphism groups?

How should I Describe automorphism groups $\operatorname{Aut}(\mathbb{Z}_9)$ and $\operatorname{Aut}(\mathbb{Z}_3 \times \mathbb{Z}_3)$ ? If a group G contains a normal subgroup H of order 9, G is ...
2
votes
2answers
115 views

List all the possible orders for G and calculate $\gamma_G(m)$

Consider a finite group G. For any integer $m \geq 1$ set $\gamma(m) = \gamma_G(m)$ to be the number of elements $g \in G$ such that ord($g$) = $m$. We say that $m$ is a "possible order" for G if ...
1
vote
2answers
313 views

Find number of abelian groups of order $27$?

I was trying to solve the following problem: Find number of abelian groups of order $27$ ? Could someone point me in the right direction? Thanks in advance for your time.