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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0answers
62 views

On primitive groups with transitive subgroups of smaller degree

Let $G$ be primitive on $\Omega$ and $G_\Delta$ transitive on $\Omega-\Delta=\Gamma$. Let $1 < |\Gamma| \le \frac{1}{2}|\Omega|$. Then $G$ is triply transitive on $\Omega$. In addition, if ...
16
votes
0answers
294 views

A question about Sylow subgroups and $C_G(x)$

Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is ...
7
votes
2answers
467 views

Computing Sylow $p$-subgroups of classical groups

Let $p>4$ be prime, and let $G=GL_2(\mathbb{F}_p)$, $H=O_3(\mathbb{F}_p)$, and $K=Sp_4(\mathbb{F}_p)$. We know that $|G|=p(p-1)^2(p+1)$, so that a Sylow $p$-subgroup of $G$ is isomorphic to ...
2
votes
1answer
86 views

Suppose that $H \leq G$ , $\phi \in Char(H)$ and $K \leq G$ that $(\phi^G)_K \in Irr(K)$. We want to prove that $G=HK$.

Suppose that $H \leq G$ , $\phi \in Char(H)$ and $K \leq G$ that $(\phi^G)_K \in Irr(K)$. We want to prove that $G=HK$. I can prove it by modules but can anybody help to prove it without using ...
1
vote
3answers
368 views

find the cyclic subgroup of $U_{21}$ generated by $[10] \in U_{21}$

Let $G = U_{21}$. Find the cyclic subgroup of $G$ generated by $[10]$. I'm not sure how to do this but here is what I tried: \begin{array}{l} g = 10 \equiv 10 \pmod {21} \\ g^2 = 100 \equiv ...
3
votes
1answer
202 views

For given prime number $p \neq 2$, construct a non-Abelian group with exponent $p$

For given prime number $p \neq 2$, construct a non-Abelian group with exponent $p$. We know that for $p=2$ it's impossible.
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0answers
53 views

A group of order 2520

Let $G$ be a group of order 2520 and let $K$ be the maximal normal soluble subgroup of $G$. If we know that $G/K\cong A_5$, $K=C_2\times (C_7 :C_3)$, $C_G(K)=SL(2,5)$ and $G= C_G(K) K$, what would be ...
6
votes
0answers
107 views

Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with 2 generators. More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of ...
8
votes
1answer
142 views

The smallest group with 3 generators

What is the smallest (in terms of the number of elements) nonabelian group such that any presentation requires at least 3 generators? Most of the nonabelian finite groups I know seem to require only 2 ...
3
votes
2answers
57 views

Is there a “natural” way to define a group operation on the set of size-$n$ subsets of a finite set?

It is easy to define a group operation on the set of all subsets of a given finite set S of size n: merely take the exclusive-or (disjoint sum) of the two sets. This is associative, the empty set is ...
2
votes
2answers
494 views

Homomorphism of a quotient group

Let $G,H$ be finite groups. Let $N$ be a normal subgroup of $G$. Let $\phi$ be a group homomorphism which maps from $G \rightarrow H$. And let $N \subset \mathrm{ker}(\phi)$ I'm trying to proof that ...
7
votes
1answer
254 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation ...
2
votes
1answer
126 views

Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$?

Let $p$ be an odd prime number and $n$ any positive integer. Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$ as groups? This seems very ...
6
votes
0answers
242 views

Examples of finite groups that are not a semidirect product

I'm looking for examples of (families of) finite groups that are not semidirect products. When first learning group theory, the first such group that one encounters is $Q_8$. In my search for other ...
1
vote
1answer
189 views

Direct product of cyclic group with itself

Let $R$ be the direct product of $C_p$ with itself. Show that $R$ is an abelian group of order $p^2$ and $R$ is not cylic.
2
votes
2answers
147 views

Subgroups of order $125$ in $S_{15}$

I know symmetric group $S_{15}$ contains a copy of $C_5 \times C_5 \times C_5$ given by generators $a=(1,2,3,4,5)$ $b=(6,7,8,9,10)$ $ c=(11,12,13,14,15)$ so $\langle a,b,c \rangle \cong C_5 ...
2
votes
2answers
91 views

Group Theoretic Correspondence: A Subtle Discussion on Function Composition Among Bijections

How is the following problem to be interpreted via the purview of group theory: Let $f$ be a one-to-one function from $X=\{1,2,\dots,n\}$ onto $X$. Let $f^k=f\circ f\circ \cdots \circ f$ denote ...
6
votes
1answer
94 views

On Decompositions of Finite Group

Any finite non-cyclic abelian group $G$ can be written as product $HK$ of two proper subgroups. Here $HK=\{ hk\colon h\in H, k\in K\}$. A step further, if $G$ is a finite group such that the ...
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vote
0answers
39 views

Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
3
votes
1answer
726 views

counting the number of elements in a conjugacy class of $S_n$

I want to know if there is some systematic way (using some combinatorial argument) to find the number of elements of conjugacy classes of $S_n$ for some given $n$. For example, let's consider $S_5$. ...
19
votes
1answer
588 views

When is $\mathfrak{S}_n \times \mathfrak{S}_m$ a subgroup of $\mathfrak{S}_p$?

Inspired by another question, I wondered when $\mathfrak{S}_n \times \mathfrak{S}_m$ is isomorphic to a subgroup of $\mathfrak{S}_p$. Eliminating the obvious cases, the question becomes: Let ...
3
votes
1answer
111 views

Number of ways a group element of a finite group can be written as a given word

I had previously asked about the number of ways a group element in a finite group could be written as a commutator (the question is still open for a proof, by the way) In how many ways can a group ...
5
votes
2answers
707 views

Computing Subgroup Lattices

Let $G$ be a finite group, and let $L(G)$ be the lattice of subgroups, partially ordered by inclusion. For example, below is $L(D_8)$. $\quad\qquad\quad\qquad\quad\quad\qquad$ I have two questions: ...
0
votes
2answers
398 views

A group of order $p^2$, where $p$ is prime, with exactly one proper subgroup is cyclic. [duplicate]

I am not able to prove this, could any one help me? $G$ be a finite group, $G$ has exactly one proper subgroup. We need to prove that $G$ is cyclic and $|G|=p^2$ where $p$ is prime. Thank you.
0
votes
0answers
76 views

Is there a subgroup of $S_{10}$ having $5040$ elements other than $S_7$?

I'm trying to generate to monomial symmetric polynomial $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 + \text{all permutations,} $$ starting from the first element by applying all ...
0
votes
1answer
217 views

the representation on the regular representation is faithful

I am reading the proof of the following proposition. Proposition. As algebras, $\mathbb{C} G \cong \bigoplus \mathrm{End}(W_i),$ where $G$ is a finite group and $W_i$ are irreducible representation ...
1
vote
1answer
117 views

A character of an induced representation

I want a help to solve the following exercise from the book, Representation Theory, by Fulton and Harris. Exercise 3.19 (p.34) Let $H$ be a subgroup of a finite group $G$. Let $W$ be a representation ...
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vote
6answers
267 views

The product of finitely many cyclic groups is cyclic

How to prove that the direct product of finitely many cyclic groups $C_{n_1}\times C_{n_2}\times\cdots\times C_{n_m}$ is cyclic if the $n_i$'s are pairwise relatively prime?
0
votes
2answers
94 views

Additive group of a finite ring of square free order is cyclic

$R$ is a finite ring of square free order $n>1$. How to show by the basis theorem for finite Abelian groups that additive group of $R$ is cyclic group of order $n$?
6
votes
0answers
265 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
0
votes
1answer
258 views

The number of irreducible representations

I am reading a textbook "Representation theory" by Fulton and Harris and I have a question. They proved the following theorem on page 16. With an Hermitian inner product on a set of class function, ...
13
votes
1answer
172 views

Is a finite group determined by the family of all its 2-generated subgroups?

At the last week I meet my old coauthor, Oleg Verbitsky who proposed me the following question. I think that here should be an easy counterexample, but I am not a pure group theorist and I am usually ...
7
votes
1answer
129 views

On $2$-groups with a property

If $G$ is a non-abelian $p$-group ($p>2$) such that any two maximal cyclic subgroups have trivial intersection, then $G$ is of exponent $p$ (see "Groups of Prime Power Order-1"- Berkovich, Exer. 2, ...
3
votes
0answers
59 views

On the Example of J. Alperin

In the paper "Large Abelian Subgroups of $p$-Groups" by J. Alperin, the author constructs an example of a $p$-group of order $p^{3n+2}$ ($p>2$) in which any abelian subgroup has order at most ...
4
votes
1answer
94 views

Can we conclude any information about the isomorphism classes of groups of order $n!$?

We know that there a isomorphism class of symmetric group structure of order $n!$. Can we conclude any other information about the isomorphism classes of groups of order $n!$ ?
4
votes
2answers
138 views

How to understand the automorphism group of a very symmetric graph (related to sylow intersections)

For a group $G$ and subgroup $H$, consider the relation on $G$ defined $x \sim y$ if $H^x \cap H^y = 1$. This defines a graph on $G$. It is always fairly symmetric: $N_G(H)$ acts on the left and $G$ ...
5
votes
1answer
144 views

Explicit isomorphism $S_4/V_4$ and $S_3$ [duplicate]

Let $S_4$ be a symmetric group on $4$ elements, $V_4$ - its subgroup, consisting of $e,(12)(34),(13)(24)$ and $(14)(23)$ (Klein four-group). $V_4$ is normal and $S_4/V_4$ if consisting of $24/4=6$ ...
10
votes
2answers
2k views

Let G be a nonabelian group of order $p^3$, where $p$ is a prime number. Prove that the center of $G$ is of order $p$.

Let G be a nonabelian group of order $p^3$, where $p$ is a prime number. Prove that the center of $G$ is of order $p$. Proof Since $G$ is not abelian, the order of its center cannot be $p^3$. ...
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vote
2answers
1k views

Show that every finite group of order n is isomorphic to a group of permutation matrices

Show that every finite group of order n is isomorphic to a group consisting of n x n permutation matrices under matrix multiplication. (A permutation matrix is one that can be obtained from an ...
7
votes
1answer
2k views

Group of order $pqr$, $p < q < r$ primes

Let $G$ be a group such that $|G|=pqr$, $p<q<r$ and $p,q,r$ are primes. i need to prove that: There exists a subgroup $H$ such that $H\unlhd G$ and $|H|=qr$. $G$ is solvable. $r$-Sylow ...
2
votes
4answers
185 views

Irreducible representation of dimension $5$ of $S_5$

i am searching for a concrete as possible description of the (there are two but the are obtained from each other by tensoring with the signature representation) irreducible representation of dimension ...
5
votes
1answer
69 views

Group of order $40$

I need to show that any group $G$ such that $|G|=40$ has 3 subgroups $H_1,H_2,H_3$ such that $H_1<H_2<H_3$ and their orders are $5,10,20$ accordingly. Thanks a million!
2
votes
2answers
330 views

If $G$ is a finite group of order $n$, why is it isomorphic to its centralizer in $S_n$?

If $G$ is a finite group of order $n$, why is it isomorphic to its centralizer in $S_n$? Here, we embed $G$ in $S_n$ via the left regular representation. From thinking a bit about the classification ...
6
votes
4answers
248 views

A group of order $8$ has a subgroup of order $4$

Let $G$ be a group of order $8$. Prove that there is a subgroup of order $4$. I know that if $G$ is cyclic then there is such a subgroup (if $G=\langle a\rangle$ then the order of $\langle ...
0
votes
1answer
81 views

$G$ be a non-nilpotent and supersoluble and 2-maximal subgroup of G permutes with all 3-maximal subgroup of G

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$, Where $P$ is a group of order $p^{2}$($p$ is prime), all maximal subgroups of $P$ are normal in $G$, $Q=\langle a\rangle$ is ...
4
votes
3answers
191 views

All distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$

This question is from a past exam. Find all distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$ Attempt/Thoughts? Since $\mathbb{Z}_4$ is cyclic we are ...
1
vote
0answers
85 views

$G$ be a non-nilpotent and supersoluble finite group.

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$ and $|Q: C_{Q}(P)|=q$ is prime, Where $P$ is a group of prime order $p\neq q$. $Q$ is noncyclic group of order ...
1
vote
3answers
290 views

the number of the subgroups of a non cyclic group whose order is $25$

My question is about group theory: How many subgroups does a non-cyclic group contain whose order is 25? How can i answer that question? Can you generalize the answer? Thanks for your help.
4
votes
3answers
79 views

How to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists?

I have 2 groups $U_5$ and $U_{12}$ , .. $U_5 = \{1,2,3,4\}, U_{12} = \{1,5,7,11\}$. I have to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists. I started with the "$yes$" case: ...
1
vote
3answers
125 views

Standard Wreath Product and Sylow Subgroups

A Sylow $p$-subgroup of $S_{p^r}$ is isomorphic with the standard Wreath Product $W(p,r) = (\cdots(C_p \wr C_p) \wr \cdots) \wr C_P)$, the number of factors being $r$. I have a great doubt as to ...