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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8
votes
2answers
273 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
3
votes
1answer
283 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
5
votes
1answer
545 views

Cyclic groups whose every non-identity member is a generator

Let $G$ be a cyclic group. There's a theorem which states that if $|G|$ is a prime, then every non-identity member of $G$ is a generator. What about a cyclic group whose order is not prime: Is there ...
2
votes
1answer
129 views

$2×2=3+1$ for $\operatorname{GL}_2$

If $V$ is the natural representation for $\operatorname{GL}(2,q)$, then $V⊗V$ appears to decompose into the direct sum of a (strange?) one-dimensional module and a three dimensional module. I've ...
14
votes
2answers
571 views

finite subgroups of PGL(3,C)

The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, ...
4
votes
8answers
534 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...
4
votes
1answer
69 views

Computation of the cokernel of the map $f: \mathbb Z \rightarrow \mathbb Z_{(2)} \oplus \mathbb Z $ defined by $f(1)=(1,2)$

$f: \mathbb Z \rightarrow \mathbb Z_{(2)} \oplus \mathbb Z $ (the sum is direct) $f(1)=(1,2)$ so the image is $\mathbb Z*(e_{1}+2e_{2})$ however computing the cokernel of this map really puzzles me ...
5
votes
1answer
472 views

Sylow subgroups

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of $G$ such that $P = (P \cap H)(P \cap K)$. It is not hard to prove that ...
14
votes
1answer
669 views

Group of even order contains an element of order 2

I am working on the following problem from group theory: If $G$ is a group of order $2n$ (i.e., the cardinality of the set $G$ is $2n$), show that the number of elements of $G$ of order $2$ is odd. ...
12
votes
2answers
446 views

Example of a group

Can one give an example of a finite group $G$, with a subset $H$ containing identity, such that $gHg^{-1}=H$ for all $g\in G$, $|H|$ divides $|G|$, but $H$ is not a subgroup of $G$. Motivation ...
12
votes
4answers
1k views

A kind of converse of Lagrange's Theorem

Let $G$ a finite group. If $a\in G$, a consequence of the Lagrange's Theorem is that the order of $a$ divides the order of $G$. Let $p=|G|$. If $p$ is prime, it is well known that $G$ is cyclic, and ...
4
votes
1answer
487 views

Exercises in Group Cohomology

I'm interested in finding a textbook to learn group cohomology, a book that contains a lot of examples and also a lot of good exercises to test my understanding. I would appreciate some feedback. ...
1
vote
2answers
66 views

For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient, if $G$ is finitely generated?

This is new variant of For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient? thanks to Alon's comment. Since that is the case I'm interested really in, I figured it's ...
6
votes
1answer
60 views

For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient?

Let $G$ be an infinite group, and $\phi$ an automorphism of it. Let $N$ be a normal subgroup of $G$ such that $G/N$ is finite. Is it true that for any $h$ in $G$, $\phi^n(h)N$ (as a sequence of ...
2
votes
2answers
179 views

Constructing the semidirect product $C_{2^n}\rtimes C_2$.

Starting from the group of automorphisms of $C_{2^{n}}$ find the automorphism $x$ of order 2 (with $n\geq 3$) and building the semidirect product $C_2$ and $C_{2^{n}}$ by $y$, where $y$ is an ...
5
votes
2answers
621 views

Show that a finite group with certain automorphism is abelian

Let $G$ be a finite group and $f:G\to G$ an isomorphism. If $f$ has no fixed points (i.e., $f(x)=x$ implies $x=e$) and if $f\circ f$ is the identity, then $G$ is abelian. (Hint: Prove that every ...
0
votes
2answers
110 views

$p$-th powers of elements of an extraspecial $p$-group

For $H$ a group and $n\in\mathbb{N}$, let $H^{(n)}=\langle h^n : h \in H \rangle$. Now let $G$ be an extraspecial $p$-group (see definition). Is it true that $G^{(p)}\cong \mathbb{Z_p}$. (It holds for ...
4
votes
2answers
263 views

is there a decomposition of $L_2(q)$ into a direct product?

I wonder if there is a nontrivial decomposition of the $L_2(q)$, where $q$ is a prime power, into a direct product. I think that there is none, but I am not sure. $L_2(q)$ refers to the special ...
24
votes
1answer
2k views

Isomorphic quotients by isomorphic normal subgroups

In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
14
votes
2answers
1k views

Isomorphic quotient groups

Suppose that G is a finite group, and that H and K are normal subgroups of G with trivial intersection, and suppose that H and K are isomorphic. Is it true that the quotient groups G/H and G/K are ...
3
votes
2answers
389 views

Explanation of claim in Dummit and Foote

Dummit and Foote, p. 204 They suppose that $G$ is simple with a subgroup of index $k = p$ or $p+1$ (for a prime $p$), and embed $G$ into $S_k$ by the action on the cosets of the subgroup. Then they ...
45
votes
3answers
917 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
4
votes
3answers
440 views

On the centres of the dihedral groups

In an proof that I recently read, the following 'fact' is used, where $D_{2n}$ denotes the dihedral group of order $2n$: If $n$ is even, then $D_{2n} \cong C_2 \times D_n$. The (short) given ...
1
vote
2answers
506 views

What are the finite subgroups of $SU_2(C)$?

Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ? What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ ...
6
votes
3answers
394 views

Transitive groups

Someone told me the only transitive subgroup of $A_6$ that contains a 3-cycle and a 5-cycle is $A_6$ itself. (1) What does it mean to be a "transitive subgroup?" I know that a transitive group ...
8
votes
3answers
379 views

Finding an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$

For an odd integer $n$, find an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$. How do I do this? I don't really know where to start. I can easily find bijections ...
10
votes
4answers
1k views

Finite groups with exactly one maximal subgroup

I was recently reading a proof in which the following property is used (and left as an exercise that I could not prove so far). Here is exactly how it is stated. Let $G$ be a finite group. Suppose ...
6
votes
2answers
727 views

Central elements in the Dihedral Group of order 2n

Continuing my independent journey through "Abstract Algebra" (see this previous question for context and notation), I am working on: If $n = 2k$ is even and $n \geq 4$, show that (a) $z = r^k$ ...
3
votes
1answer
2k views

Computing the order of elements in Dihedral Groups

I am working through "Abstract Algebra" by Dummit & Foote. Exercise 1.2.1 states: Compute the order of each of the elements in ... $D_6$, $D_8$, and $D_{12}$. I have found that: ...
7
votes
1answer
273 views

Explanation why an abelian tower admits a cyclic refinement

Now that school is wrapping up, I'm trying to crack down and get better at algebra. This proposition from Lang's Algebra loses me at the end. Here is my understanding so far: (Please excuse me if a ...
3
votes
1answer
73 views

$\vert G \vert < \infty$, $A,B,C \leq G$, $B \leq A \Rightarrow \vert A:B \vert \geq \vert C \cap A : C \cap B \vert$

I am stuck with the following problem. I am sure it cannot be that hard since it is intuitively true, but I can't find a way to prove it. Let $A,B,C \leq G$ where $G$ is a finite group. Suppose ...
5
votes
1answer
177 views

Centralizer of a $p$-element modulo the $p'$-core and conjugacy class sizes in quotient groups

Does $[ G : C_G(x) ] = [ G/K : C_{G/K}(x) ] [ K : C_K(x) ]$ hold for all finite groups $G$ and $p$-elements $x$, where $K = O_{p'}(G)$ is the largest normal subgroup of $G$ with order coprime ...
7
votes
3answers
671 views

Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$

I need help with the following problem: Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
4
votes
1answer
237 views

Wikipedia article on Sylow's theorems

(Edit: The wikipedia article is correct. I messed up the notions normalizer and normal closure.) Please take a look at the wikipedia article on Sylow's theorems here, more precisely, at the last ...
3
votes
2answers
145 views

$B,N,H$ be subgroups of $G$, is it true that $ \langle B \cap H, N \cap H \rangle = \langle B,N \rangle \cap H $?

Let $G$ be a finite group, let $B,N,H$ be subgroups of $G$. I believe that $$ \langle B \cap H, N \cap H \rangle = \langle B,N \rangle \cap H $$ but I do not find a satisfactory proof. I think this ...
10
votes
4answers
1k views

Index of a maximal subgroup in a finite group

My question is very simple and maybe trivial. Here it is. Is the index of a maximal subgroup in a finite group always a prime number? Thanks in advance.
1
vote
2answers
195 views

Groups/Linear maps

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
11
votes
2answers
942 views

Generalization of index 2 subgroups are normal

Let $G$ be a finite group and $H$ a subgroup of index $p$, where $p$ is a prime. If $\operatorname{gcd}(|H|, p-1)=1$, then $H$ must be normal. Does somebody have a quick proof of this?
4
votes
1answer
271 views

Semidirect Products of the form $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$

What are the different (non-isomorphic) semidirect products $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$, when $\phi \colon ...
6
votes
1answer
433 views

Symmetries of Cube

The group of orientation preserving isometries of Cube is $S_4$. But if we allow orientation reversing isometries also, then the group will be of order 48. What is this group (Structure)? ( Part of ...
4
votes
2answers
452 views

Subgroups whose order is relatively prime to the index of another subgroup

Suppose that $H, K$ are subgroups of a finite group $G$, with $|H|$ relatively prime to $|G:K|$. Does it necessarily follow that $H \leq K$, or is there a counterexample? This question arose from ...
12
votes
3answers
656 views

Dimensions of irreducible representations of finite groups over $\mathbb Q$

If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the ...
2
votes
2answers
180 views

Sylow $5$-subgroup in $S_{16}$

Find two different $5$-Sylow subgroups in $S_{16}$. Hint: use group multiplication. Any hints?
11
votes
1answer
345 views

Normalizing every Sylow p-subgroup versus centralizing every Sylow p-subgroup

Is it true that: If the intersection of the Sylow p-subgroups is trivial, then the intersection of their normalizers is equal to the intersection of their centralizers? I half remember this ...
2
votes
0answers
204 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
5
votes
3answers
813 views

Group of order 12

Is it true or false that a group of order 12 always has a normal 2-sylow subgroup? I have a hunch it is false..
7
votes
3answers
300 views

Finite groups: $H \leq A \times B$. Is $H \cong C \times D$ for some $C \leq A$, $D \leq B$?

$A$ and $B$ are finite groups. $H \leq A \times B$. Can we find some $C \leq A$, $D \leq B$ such that $H \cong C \times D$? In case the statement is not true: is it true under further assumptions ...
20
votes
7answers
2k views

Product of all elements in an odd finite abelian group is 1

This should be an easy exercise: Given a finite odd abelian group $G$, prove that $\prod_{g\in G}g=e$. Indeed, using Lagrange's theorem this is trivial: There is no element of order 2 (since the order ...
7
votes
3answers
2k views

A normal subgroup intersects the center of the $p$-group nontrivially

If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
4
votes
1answer
441 views

Find a subgroup of the octic group that is normal, and one that is not normal

Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$. Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$. Find a subgroup of $G$ that has ...