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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46 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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0answers
19 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
1
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1answer
22 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
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0answers
13 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
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1answer
25 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
1
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1answer
26 views

number of generators of a semi direct product

Let $G$ be a finite group. Let $g(G)$ be the minimum set of elements of $G$ required to generate the whole group. Suppose that $G= H \rtimes K$ is a semi direct product of two finitely generated ...
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0answers
18 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
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2answers
892 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
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1answer
58 views

When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ...
4
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1answer
2k views

Existence of normal subgroups for a group of order $36$

Prove that a group of order 36 must have a normal subgroup of order 3 or 9. Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then n must be 1 or 3. Let n3 be the number of 3-Sylow ...
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0answers
19 views

Proving a conjugation map is an Inner automorphism of a group

Definition: The map $i_{g}:G\rightarrow G$ $h\mapsto g^{-1}hg$ Lemma: $i_{g}$ is an Automorphism of G called an Inner Automorphism. My attempt to prove this is as follows: ...
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1answer
61 views

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$.

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$. I think this is associated with the action of the left coset of ...
2
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1answer
42 views

Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why?

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$. Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...
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0answers
23 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
1
vote
1answer
58 views

What is the next term in the jumping champions?

I am interested in the jumping champions of the number of groups of order $n$, where $n$ is cubefree. After calling LoadPackage("cubefree"); GAP gives the ...
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0answers
19 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
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1answer
21 views

The minimal group with Fitting length three has $p$ section in middle?

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. Assume that for every proper characteristic subgroup $K$ of $G$, Fitting ...
0
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0answers
11 views

Exponent operation over elements of $G$

I found a definition of an exponent operation over the element of $\mathbb{G}$ in this paper (page 4): $$ (g^a)^{\% b} = g ^{a \text{ mod } b}$$ I couldn't understand the rest of the paper (Decrypt ...
2
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1answer
57 views

Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order ...
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1answer
38 views

Can every group be obtained from a choice of Sylow subgroup for every prime divisor?

The question is almost clear from title: If $G$ is a finite group of order $p_1^{n_1}\cdots p_r^{n_r}$ then is it always possible to choose one Sylow subgroup for every prime divisor of $|G|$ ...
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0answers
37 views

Proof of Cayley's Theorem

This question relates to the link: Cayley's theorem The way I reasoned in showing the map T is a Homomorphism is as follows: Definition: A Homomorphism $\phi: \left ( G,\ast \right ) ...
4
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2answers
93 views

What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such ...
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2answers
43 views

Automorphism of a group is a group action [on hold]

Let G be a group and let $\Omega$ be a set. Then, the $Aut\left ( G \right )$ acts on $\Omega=G$ How can I show that this is true? Thank in advance.
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3answers
41 views

A certain centralizer group in finite group of order $3^5$ of maximal class contains every normal subgroup of index greater than $3$

Let $G$ be a finite group of order $3^5$ and maximal class, i.e. if $K_{i+1}(G) = [K_i(G), G]$ denotes the subgroups of the lower central series, we have $K_5(G) = 1$ and $K_4(G) = Z(G)$. Set $G_1 = ...
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0answers
38 views

unfaithful group action [on hold]

Let the group $G=GL\left ( n,\mathbb{F} \right )$ and $\Omega$ be the set of all 1-dimensional subspace of $\mathbb{F}^{n}$ Let $\left \langle V \right \rangle \in \Omega$. Define $\left( \langle ...
0
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1answer
23 views

An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
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0answers
27 views

About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
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1answer
50 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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0answers
22 views

references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
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1answer
17 views

Intersection of two subgroups with given information

suppose we know that $G$ is a finite group with order $43200$ and suppose that $H$ is a subgroup of $G$ with order $80$. Furthermore, assume that $K$ is also a subgroup of $G$ such that $[G:K]=1600$. ...
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1answer
24 views

showing a set is a subgroup of a normaliser

Let $H$ be a subgroup of a group $G$ and defined $N_{G}\left ( H \right )=\left \{ g \in G \mid g^{-1}Hg=H \space\ \right \}$ Show that $H$ is a normal subgroup of $N_{G}\left ( H \right ).$ The ...
0
votes
1answer
6 views

Specific condition for a map to be isomorphism

Let $G=\left ( \mathbb{R} \space\ \text{where} 0 \notin \mathbb{R},\cdot \right )$ and let r be a positive integer. Define $\phi:G\rightarrow G$ $x \mapsto x^{r}$ Show that $\phi$ is an ...
0
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1answer
15 views

Order of an element in an external direct product

Consider $\mathbb{Z}_{4}\times \mathbb{Z}_{4}=\left \{ 0,1,2,3 \right \}\times \left \{ 0,1,2,3 \right \}$ The element $\left ( 2,0 \right )$ is of order 2 but I cannot figure out why. $2=LCM\left ...
4
votes
3answers
130 views

If $G$ is a finite group and $H \subset G$ is closed, must $H$ be a subgroup?

Supposed theorem (from online notes): If $(G,*)$ is a finite group and $H\subset G$, $H$ is non-empty and $H$ is closed under $*$ then $(H,*)$ is a group. The proof given is a real mess, but, after ...
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0answers
49 views

Upto which prime is the calculation of $gnu(2^4\cdot3^4\cdot…\cdot p^4)$ feasible?

The determination of $gnu(n)$, the number of groups of order $n$ upto isomorphism, is very hard in general. But if no large powers are involved, it should be possible for relatively large numbers. ...
4
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1answer
58 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
0
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1answer
39 views

finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6

If I have a finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6, is there anything special about G that we can infer? Would the order of $G$ be 30?
2
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2answers
22 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
3
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1answer
166 views

What exactly is a p'-group?

In the context of finite group theory I understand that $p$-group is a group whose order is a power of $p$ ($p$ a prime number) but I am unclear on the exact definition of a $p'$-group. This ...
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0answers
40 views

Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
7
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3answers
1k views

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements. We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$. If $G$ is ...
3
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1answer
45 views

Is the following equation equivalent to the 2-cocycle condition?

Given a finite abelian group $G$, I'm looking for functions $\rho:G \times G \to U(1)$ such that 1) $~~~~~\rho(g,e) = 1 = \rho(e,g)$, where $e\in G$ is the unit element and such that 2) ...
3
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1answer
41 views
2
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1answer
46 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow ...
1
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1answer
122 views

If a finite group acts transitively on a set, does its center also acts transitively? [closed]

If $G$ is a finite group acts transitively on a set $X$, does the center $Z(G)$ also acts on $X$ transitively? I don't see how this statement will be true but I can't come up with a counter ...
2
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0answers
230 views

Results on Hall $\pi$-subgroups - proof verification

I would appreciate verification of the following proofs. The problems are all closely related and the proofs are similar, so I think it's only appropriate to post them together (also, they are part ...
2
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1answer
54 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
0
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0answers
53 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
0
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1answer
25 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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0answers
14 views

Vertex-transitivity of the automorphism group of a digraph

I am trying to understand the theorem 3 of Cycles in graphs and groups by Kantor. Theorem $3$ If $G$ is a vertex-transitive group of automorphisms of a digraph $\Gamma$ with outdegree $d \ge 1$, ...