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Questions tagged [finite-groups]

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.

3
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0answers
44 views

Exponent of the group $GL(n,\mathbb F_2)$

Let $A\in GL_n(\mathbb F_2)$ be an element of order greater than or equal to $2^n-1$ . Then is it true that order of $A$ is $2^n-1$ ? I know that $|GL_n(\mathbb F_2)|=(2^n-1)(2^n-2^2)...(2^n-2^{n-1})$...
0
votes
1answer
24 views

Some results given that $|G| = nm$

Let $G$ be a finite group such that $|G| = nm$. Suppose $x\in G$ has order $n$ and let $\sigma_x\in S_G$ be the permutation such that $\sigma_x(g)=xg$ for every $g\in G$. Note that $\sigma_x$ is a ...
1
vote
1answer
18 views

Understanding conjugacy classes in $SL_{2}(\mathbb{F}_{q})$

I am trying to calculate the conjugacy classes of the group $SL_{2}(\mathbb{F}_{q})$, with the help of the knowledge of conjugacy classes of $GL_{2}(\mathbb{F}_{q})$. I am using two of the following ...
1
vote
1answer
29 views

Show that $\operatorname{GL}_2 (\mathbb{F}_3)/\{\pm I_2\} \cong S_4$

Let $\mathbb{F}_3$ be a field with three elements and let $V = \mathbb{F}_3^2$. Let $\alpha,\beta,\gamma$ and $\delta$ be the four one-dimensional subspaces spanned by $\begin{bmatrix}1\\0 \end{...
1
vote
2answers
24 views

Standard representation of $S_n$ is irreducible

Let $S_n$ act on an $n$-dimensional $\mathbb{Q}$-vector $V$ space with basis $\{v_1,\cdots,v_n\}$ by $\sigma(v_i)=v_{\sigma(i)}$. Consider subspaces $W_1=\langle v_1+v_2+\cdots + v_n\rangle$ and $W_2=...
0
votes
0answers
32 views

Question from Algebra: Chapter 0 by Aluffi [duplicate]

Let K be a normal subgroup in finite group G. Assume |K| and [G:K] are relatively prime. Is K characteristic in G? I am self-studying through this book. Please only provide very small hints.
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0answers
34 views

Help proving Sylow's Theorem order 58

Prove that every group of order 58 is not simple. So I know that 58 = 2 ⋅ 29. I assume G is simple. I'm having trouble using the Sylow Theorems to show that this is not Simple. In particular, ...
0
votes
1answer
16 views

Proving that if $G$ is a group of order $p^2$ where $p$ is prime, then $G$ is either $\mathbb{Z}_p \times \mathbb{Z}_p$ or $\mathbb{Z}_p$

Prove that if $G$ is a group of order $p^2$ where $p$ is prime, then $G$ is either $\mathbb{Z}_p \times \mathbb{Z}_p$ or $\mathbb{Z}_{p^2}$ I'm aware that this question has an answer here : There ...
1
vote
0answers
35 views

Identifying result from a Cayley graph in gap

I constructed a group s which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ using GAP commands as below. ...
3
votes
1answer
36 views

“Tensor complement” of representations of finite groups: reloaded

Let $V$ be a finite dimensional simple $G$-representation (over $\mathbb{C}$) for a finite group $G$. Let $R$ be the regular representation of $G$. Is there a $G$-representation $W$ and $k\geq 1$ ...
6
votes
1answer
33 views

Sum of values of an irreducible character is non-negative integer

I am trying to prove the following fact: If $G$ is a finite group, $\chi$ is a complex irreducible character of $G$ and $\{g_1,\cdots,g_r\}$ is a complete set of representatives of conjugacy ...
5
votes
1answer
33 views

Let $G$ be a finite matrix group in $GL_2(Q)$ such that every matrix $A\in G$ has integer entries. Prove $A^{12}= I$ for each $A$.

Let $G$ be a finite matrix group in $GL_2(Q)$ (general linear group of $2$ by $2$ matrices with rational entries) such that every matrix $A\in G$ has integer entries. Prove that $A^{12} = I$ for every ...
3
votes
1answer
40 views

If $S$ is subnormal in $G$, $S$ is simple and nonabelian and $S \subseteq H \subseteq G$, then $S \subseteq \textrm{Soc}(H)$.

I suspect I have missed some easier way to show this claim, and there might be a mistake in my approach. I know this is a very lengthy proof, but I spent a lot of time on trying to solve this problem ...
1
vote
1answer
25 views

Finite order elements commutes with group

Suppose group $G$ has only finitely many elements with finite order, call this torsion subset $H$. Then there exists $n$ such that $a^n b=ba^n$ for all $a\in G$ and $b\in H$. Question is how to show ...
-2
votes
0answers
33 views

Does the “Group Explorer” software help with visualizing groups? [on hold]

Does anyone know whether we can visualize the groups that we want by using the software called "Group Explorer" (SourceForge link)? I mean can we take direct products or semidirect products of groups ...
1
vote
0answers
41 views

Classifying elements of finite and infinite order in $GL_{2}(\mathbb{R})$

A problem on a recent assignment defined the Torsion subset $F(G)$ of a Group $G$ as the set of elements of G of finite order. It then asked to prove that $F(GL_2(\mathbb{R}))$ is not a subgroup of $...
0
votes
0answers
42 views

How many (non-isomorphic) abelian groups of order $200$ are there?

I used the fundamental theorem of finite abelian groups. $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5}$ $\mathbb{Z}_{2} \times \mathbb{Z}_{2}...
0
votes
0answers
17 views

Problem related to semidirect product

I have a small question regarding the semidirect product. Consider a group $G$ which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ (internal semidirect product). ...
1
vote
0answers
50 views

why there is no non cyclic group other than $D_{61}$. [duplicate]

I know that there are two non isomorphic groups of order $122$. One is cyclic group of order $122 $ and another is $D_{61}$. I know that all cyclic groups are isomorphic. But I can not understand why ...
1
vote
1answer
32 views

Describe the subgroups of S_5 generated by the 5-cycles

I'm new to Group theory and I'm just checking on my understanding. One example of 5-cycle is $(1\ 2\ 3\ 4\ 5)$. Hence, a subgroup generated by this 5-cycle consist of $\{(1\ 2\ 3\ 4\ 5), (1\ 3\ 5\ 2\ ...
1
vote
1answer
30 views

Minimal generating set for product of groups

Let $S_1$ be a minimal generating set for a group $G_1$, $S_2$ be a minimal generating set for a group $G_2$. $T_j:G_j\to G_1\times G_2, j=1,2 $. Then can we say $T(S_1) \cup T(S_2)$ is a "minimal" ...
0
votes
0answers
21 views

What is the exponent of a finite abelian group?

If $G$ is an abelian finite group and a direct product of cyclic that's order power of primitive, what is the exponent of $G$? It's right that its $p$-group and the exponent is the $lcm$ of all orders ...
0
votes
0answers
23 views

Question regarding Cayley graphs and digraphs

If a hamiltonian cycle exist in the Cayley digraph of a particular group then can we say a hamiltonian cycle exists for the Cayley graph(undirected) of the same group?
0
votes
1answer
53 views

Prove That if |a|=m and |b|=n and ⟨a⟩∩⟨b⟩={e} then, gcd(m, n)=1 [closed]

$G$ is a group and $a,b,\in G$. To summarize the question, if the cyclic group generated by $a$ and $b$ only has the identity element in common, then the orders of $a$ and $b$ are relatively prime. ...
0
votes
0answers
20 views

Request for clarification about a computation in semidirect products

In my question, For group G which is the Semidirect product of $\mathbb{Z}_3$ and $\mathbb{Z}_7 \times \mathbb{Z}_7$, where $\mathbb{Z}_7 \times \mathbb{Z}_7$ is the normal subgroup, if z is a ...
-2
votes
1answer
48 views

How do I prove that $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},e^{2\pi i/3})/\mathbb{Q})$ is isomorphic to $S_{3}$?

I know how to obtain the automorphisms, but I am not able to prove the isomorphism between both groups. Can someone help me? Thanks in advance.
0
votes
0answers
37 views

Understanding the Great Orthogonality Theorem (group theory)

I want to build some intuition around the important Great Orthogonality theorem. The Great Orthogonality theorem states that for a finite group $G$, we have (in a particular form of the theorem): $$...
0
votes
1answer
35 views

What is the intuition behind the direct product and the direct sum of groups being identical for finite groups?

I have seen the statement that the direct product of groups $G_1\otimes G_2 \otimes … \otimes G_N$ is identical to the direct sum of groups $G_1\oplus G_2 \oplus … \oplus G_N$ for finite $N$. What ...
1
vote
0answers
43 views

How do I find out which Galois groups are isomorphic to some given groups?

More generally, I would also like to know which Galois Groups are isomorphic to $\mathbb{Z}/3$, $\mathbb{Z}/5$, $\mathbb{Z}/7$, $\mathbb{Z}/9$ and $\mathbb{Z}/3\times\mathbb{Z}/3$ as well as the ...
0
votes
1answer
26 views

Prove that a finite cyclic group of order $n>2$ has an even number of distinct generators. What can you deduce about $\phi(n)$ when $n>2$?

I have not yet learned any formulas to compute $\phi(n)$ (Euler phi function), nor am I familiar with its properties. As such, I currently do not have the tools to prove directly that $\phi(n)$ is ...
0
votes
1answer
42 views

Isomorphism from $U(st) →U(s)\oplus U(t)$

Let $s,t$ are relatively prime then $U(st) $ is isomorphic to $ U(s)\oplus U(t)$. Define a function from $U(st) →U(s)\oplus U(t)$ by $x\rightarrow (x $mod $s,x$ mod $t)$. I proved this is 1-1 ...
6
votes
2answers
218 views

Is it possible to demonstrate two groups are isomorphic without specifying an isomorphism between them?

Usually, when there are two finite groups of small order, we can check if they are isomorphic by trying to impose an isomorphism. But suppose we have two very large finite groups, what are some ...
-1
votes
3answers
49 views

Examples of groups that are not rings

I am teaching some really advanced high school students about groups and rings and wondering of examples of groups that are not rings. I am hoping to find such examples where addition and ...
1
vote
1answer
22 views

Are these two semidirect products isomorphic?

Let $p$ be a prime. Then there is a nonabelian semidirect product of $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p$. There is also a nonabelian semidirect product of $\mathbb{Z}_p\oplus\mathbb{Z}_p$ and $\...
3
votes
1answer
30 views

When a simple group normalizes a subnormal subgroup of a finite group

Out of curiosity I am working my way through Isaacs's Finite Group Theory and am stuck on problem 2A.4: Let $(G,*)$ be a finite group with simple subgroup $N$ such that $\forall H \lhd\lhd G: ...
0
votes
1answer
46 views

Action inside semidirect products

When considering a group G which is the Semidirect product of $\mathbb{Z}_3$ and $\mathbb{Z}_7 \times \mathbb{Z}_7$, where $\mathbb{Z}_7 \times \mathbb{Z}_7$ is the normal subgroup, if z is a ...
1
vote
1answer
59 views

Show that $P$ is a Sylow $p$-subgroup of $G$ if and only if $P$ is a Sylow $p$-subgroup of $N_G(P).$

Question: Let $G$ be a finite group and $P$ be a $p$-subgroup of $G.$ Show that $P$ is a Sylow $p$-subgroup of $G$ if and only if $P$ is a Sylow $p$-subgroup of $N_G(P).$ My attempt: Suppose that $|...
0
votes
1answer
29 views

using euler's theorem (phi/totient function) to compute order of group elements

The question is to prove that every element of $(Z / 72Z)^{\times}$ has order dividing $12$, somehow using Euler's theorem to first reach the fact that every element has order dividing $24$, and then ...
2
votes
1answer
28 views

Socle of a direct product of finite groups.

The socle of a group $G$ is defined as the subgroup generated by minimal subgroups among normal subgroups of $G$, and it is denoted as $\textrm{Soc}(G)$. Suppose $A_1,...,A_n$ are finite groups. Is ...
1
vote
1answer
49 views

Question regarding the notations in GAP output

I obtained the group $G$ which is the semidirect product $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ in GAP as below. ...
0
votes
1answer
15 views

Divisibility of $\operatorname{ord}\left(b\right)$ by specific power of $r$ for $b \in \mathbb{F}_q \setminus \{0\}$ where $r$ prime

I am currently studying toward understanding the Berlekamp–Zassenhaus algorithm for factorizing polynomials, using 'Algorithms of Informatics' (AnTonCom, Budapest, 2011). While studying some algebraic ...
1
vote
1answer
28 views

If $k(G) > \frac{\vert G\vert}{p}$, prove that $Z(G) \neq 1$

Let $G$ be a non-trivial finite group and let $p$ be the least prime divisor of $\vert G\vert $. If $k(G) > \frac{\vert G\vert}{p}$, prove that $Z(G) \neq 1$. Suppose for a contradiction that $...
1
vote
0answers
21 views

Equivalent Definitions of Finite Groups of Lie type

I am reading Carter's book on finite groups of lie type and am looking for a reference, or maybe a hint on the following statement in the introduction. He defines a finite group of lie type as the ...
0
votes
0answers
26 views

Question regarding factor group lemma

Factor Group Lemma: Suppose that 1.$N$ is a cyclic, normal subgroup of group $G$. 2.$(s_1,s_2,\ldots,s_m)$ is a hamiltonian cycle in $Cay(G/N;S)$. 3.The product $s_1s_2\cdots s_m$ generates $N$. ...
0
votes
0answers
49 views

The automorphism group of a finite group

Let $G$ be a finite group. Suppose $G=M\times M\times \cdots\times M$ where $M$ is a non-abelian simple group. Question: 1) What is the automorphism group of $G$? 2) What is the inner ...
0
votes
1answer
31 views

Left & Right Cosets of $A_{4}$ in $S_{4}$

So I'm looking through solutions for a practice problem in my modern algebra textbook that has me completely stumped. Let $S_{4}$ be the group of permutations of the set $\{ 1 , 2 , 3 , 4 \}$, and ...
0
votes
0answers
16 views
2
votes
2answers
72 views

Can graphs in GAP be obtained as a visual output

In GAP when we draw a cayley graph of a group using "CayleyGraph" command we get a list. Is there a way to visualize that cayley graph like a figure? Or else is there any other software that has the ...
2
votes
1answer
30 views

Write $\mathbb{Z}_{20}^{\times}$ as a product of p-power cyclic groups.

Can anyone let me know if my answer is okay? Write $\mathbb{Z}_{20}^{\times}$ as a product of p-power cyclic groups. I showed that in general $\mathbb{Z}_{n}^{\times}=U(n)\leq \mathbb{Z}_{n}$, and ...
1
vote
1answer
22 views

Can I find a series where an abelian series of smallest possible length is different from derived series?

https://groupprops.subwiki.org/wiki/Derived_length Here I have found two definitions of derived length. How to prove the equivalency of these two definitions. I know that derived series slows down ...