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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10
votes
1answer
141 views

Prove $G$ is abelian if $f(f(x)) = x$?

Let $G$ be a finite group and $f$ an automorphism such that $f(f(x)) = x$, and $f(x) = x$ if and only if $x=e$. Prove that $G$ is abelian and $f(x) = x^{-1}$. My attempt: ...
2
votes
1answer
67 views

How to find group homomorphisms from one group to another

I am trying to figure out all the homomorphisms from $\mathbb{Z}_2\times\mathbb{Z}_2$ to $\mathbb{Z}_2$. Is there a good process for doing such a think? I am getting lost...
0
votes
1answer
48 views

Quotient group of a direct product: $\left(G_1\times G_2\right)/\left(G_1\times \{e_2\}\right)$

Let $G_1$ and $G_2$ be groups, and $G := G_1 \times G_2$. Let $e_2$ be the identity element in $G_2$. Show that $H := G_1 \times {e_2}$ is a normal subgroup of G. Using the homomorphism ...
1
vote
1answer
58 views

How is this automorphism an inversion? And how is this group abelian?

Let $G$ be a finite group, and let $T$ be an automorphism of $G$ which sends more than three-quarters of the elements of $G$ onto their inverses. Then how to demonstrate that $T(x) = x^{-1}$ for all ...
3
votes
4answers
214 views

General approach for finding how many group homomorphisms are there

So I've asked this type of questions for more than once, and still I don't get the method(s) I've been presented with. What's the general recommended method for finding how many homomorphisms are ...
4
votes
1answer
363 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
2
votes
2answers
41 views

Average number of distinct values

Let $q$ such that $q < n$. I pick at random $n$ values in $\mathbb{F}_q$. What is the average number of distinct values ? Thank you
1
vote
1answer
45 views

Modular arithmetic and maximal permutations

I have a research paper about pseudo-random number generators and I need to answer the following: Given $n \in \mathbb{N}$, let's consider the permutation group of $A=\{{0,1,\dots,n-1}\}$. Since $A$ ...
3
votes
0answers
84 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
5
votes
2answers
136 views

Generality of rings' abelian group

Let G be an abelian (finite) group. Is there a ring $R$ with $G$ isomorphic to the group $(R,+)$?
1
vote
0answers
72 views

How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
6
votes
2answers
117 views

Rank of $(G/H)/(G/H)_t$ where $G$ is finitely generated abelian and $H$ is a subgroup.

Let $G$ be a finitely generated abelian group and $H$ be a subgroup. Let subscript $t$ denote the torsion subgroup. If $G/G_t$ is free of rank $n$ and $H/H_t$ is free of rank $m$, it is easy to embed ...
5
votes
1answer
102 views

Number of subgroups of order $4$ and $8$ in a group of order $72$

Let $G$ be a group of order $72$. I want to calculate the number of subgroups of order $4$ and $8$ with GAP. How can I do? thanks in advance.
2
votes
1answer
45 views

Automorphism of group of $\mathbb{Z}_p^{\oplus n}$

Let $\mathbb{Z}_p^{\oplus n}$ be the direct sum of $n$ copies of $\mathbb{Z}_p$. How do I see that the group of group automorphisms Aut$(\mathbb{Z}_p^{\oplus n})$ has the same order with the group ...
0
votes
0answers
28 views

Is this sort of a representation possible in a group with these conditions? [duplicate]

Let $G$ be a finite group, and let $T$ be an automorphism of $G$ such that $T(x) = x$ for $x \in G$ if and only if $x = e$. Then is it possible to represent every element $g \in G$ as follows? $g = ...
2
votes
2answers
232 views

$\langle x \rangle$ is a direct summand of a finite abelian group where $x$ is maximal order [duplicate]

Let $x$ be an element of a finite abelian group $G$ where $x$ has maximal order. Then I want to show that $\langle x\rangle$ is a direct summand of $G$. Note that I do not want to use finite abelian ...
0
votes
1answer
91 views

Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
1
vote
1answer
78 views

Orbit-Stabilizer Theorem proofs?

I posted 3 full questions to give context. But my main problem is the second part of the questions and how would they be answered/proved.
0
votes
2answers
121 views

$\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
4
votes
1answer
225 views

How to find presentation of a group using GAP?

I have a group from Small Group Library and I want to find its presentation using GAP. I have tried to use PresentationFpGroup(G) but failed. Please suggest me a method.
0
votes
1answer
40 views

Why is $U(12) = U_{4} (12) ~ U_3(12)?$

Why is $U(12) = U_{4} (2)~ U_3(12)$ Attempt: Any subgroup $U_k(n) = \{x \in U(n)~~|~~x \mod k=1 , k ~|~n \}$ Hence : if $U(12) =\{1,5,7,11\}$ then : $U_4(12) =\{1,5\}$ and $U_3(12)=\{1,7\}$ We see ...
1
vote
2answers
133 views

Sylow $7$-subgroup of a group of order $4\cdot3\cdot5\cdot7$ is normal

How to show that the sylow $7$-subgroup of a group of order $420$ is normal. I Know that it is true using GAP. But how to show it without using GAP. I don't know how to start this. Thanks for any ...
1
vote
1answer
55 views

Upper bound of the order of a group generated by 2 elements with certain relations

Let $G$ be generated by ${a,b}$ with relations $a^8 = b^2a^4 = ab^{-1}ab = e$. I want to show that $G$ has order at most 16. What's a good way of solving problems of this kind? Let $N$ be the normal ...
0
votes
1answer
125 views

Proving that Aut($S_3$) is isomorphic to $S_3$

I'm doing an exercise were I had to first prove that all automorphisms of $S_3$ induce a permutation in $X= \{ \alpha \in S_3 \, / \, $order$(\alpha) = 2\}$, which was easy enough. Now I have to ...
1
vote
1answer
57 views

Uniqueness FTOFAG

How do you prove uniqueness for the fundamental theorem of finite abelian groups? The book I'm using has this not very well written proof that I can't follow. So following this proof, I multiply by ...
0
votes
0answers
71 views

Boosting of Coset diagrams

If we have the diagram that represents a transitive permutation representation of $(p,q,r_o)$ for some $p, q$ and $r_o$, we often use this diagram to get diagrams for any $r>r_o$. We can do this ...
0
votes
1answer
92 views

Symmetry Group Regular Tetrahedron

Looking for some help of how to do this, which could also be expanded to other shapes. Thanks.
2
votes
6answers
264 views

Example of not so simple group ??

Help me with an example of a group having subgroups but it doesn't admit a normal subgroup.. ?? The alternate definition of simple groups using non trivial homomorphic image. Searching for a map that ...
0
votes
1answer
75 views

definition of simple group , why we need normal ???

Every prime ordered group is simple, its because it doesn't admit any subgroups. But where comes the normal subgroup, why cant the people use just subgroups instead of normal subgroups in the ...
2
votes
1answer
70 views

On the Hall Theorem

I went to search info about it in books (as Isaac), this site, wikipedia. But no one says nothing about my (maybe silly) doubt. The theorem states that, given a finite and solvable group $G$, and an ...
1
vote
0answers
51 views

$|G':G''| \le p^2$ implies $G'$ is abelian [duplicate]

If $G$ is a finite $p$-group, and $|G':G''| \le p^2$, then $G'$ is an abelian group. I'm reading its proof but I cannot understand a part: Suppose $G''\neq1$, then by a theorem, there exists a ...
2
votes
1answer
90 views

Cayley's Theorem - Questions on Proof Blueprint [Fraleigh p. 82 theorem 8.16]

Not a duplicate of this exquisite answer, the numbers in which I abide by here. Not querying the proof, hence please don't discourse on it. Proof blueprint: Steps 1-2 in words. Left multiplication ...
0
votes
1answer
57 views

Group Theory, Permutations.

Let $x := (175)(2436)$ and $y := (1234567)$ Compute the elements $x^{-1}yx$ and $y^{-1}xy$? And if $z := (1326745)$ for which $u$ is there an element $v$ such that $v^{-1}zv=u$. If $u$ exists find ...
1
vote
2answers
27 views

if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
1
vote
1answer
73 views

How to count generators for a cyclic group

Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
1
vote
4answers
78 views

If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

Hint: Use Fermat's Theorem that $a^{p-1}\equiv 1\mod p$ if $p \nmid a$. (I have no idea, but something in group theory should help)
0
votes
1answer
37 views

Surjective Homomorphic map and orders

I have recently taken a test and this question gave me a problem and left me confused and unsure how to answer. I did answer it although I'm pretty sure I didn't get it right. So I'm asking for help ...
4
votes
1answer
232 views

Subgroups of a finite elementary abelian group.

I am looking for a method to calculate number all subgroups of a finite elementary abelian $p$-group. Suppose $G$ be an elementary abelian $p$-group of order $p^n$. A proper subgroup $H$ of $G$ is ...
1
vote
2answers
190 views

The Proof of Wilson's Theorem using the auxiliary multiplicative modulous group [duplicate]

(self answered question, thanks for the hints Derek Holt provided:-)) problem 18,section 4 chapter 2 in Herstein's abstract algebra: Using the results of Problem 15 and 16,prove that if p is an ...
2
votes
1answer
52 views

A question about direct products of subgroups of a finite group.

Suppose $H$ and $K$ are subgroups of a finite group $G$ where $|H||K|=|G|$. Show that $H\cap K=\{e\}$ iff $G=HK$ $\rightarrow$ Suppose $|H|=m$. Let $H=\{h_0,h_1,h_2,....,h_{m-1}\}$. Since ...
0
votes
1answer
30 views

What is the possible value for the subgroup of index 2?

Let $H\leq G$ and $[G:H]\leq 2$. If the $|G|=n$ then $|H|=n$ if $n$ is odd. What is happening when $n$ is even? What are the possible values for $|H|$ when $n$ is even? Thanks.
2
votes
3answers
187 views

If G is a finite abelian group and $a_1,…,a_n$ are all its elements, show that $x=a_1a_2a_3…a_n$must satisfy $x^2=e$.

I have already tried with $S_3$, and indeed, the product is $(13)$, and $(13)^2=e$ But what about this: I define + in this way:$45=2$,$26=3$, $1$ is the identity. therefore, $123456=2326=233=3$, ...
1
vote
1answer
110 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
0
votes
2answers
67 views

solutions of $x^2\equiv 1 \pmod p $ [duplicate]

If p is a prime, show that the only solutions of $x^2\equiv 1 \pmod p $ are $x=1$ and $x\equiv -1 \pmod p$. (from herstein's abstract algebra chapter2 section4 lagrange's theorem problem 15, this ...
0
votes
0answers
32 views

Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k $ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
0
votes
1answer
21 views

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
0
votes
1answer
24 views

Evidence about the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$

Be $p$ an odd prime number. Show that the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$ has a unique element of order $2$, namely $\overline{p-1}$, and show that ...
1
vote
1answer
28 views

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$, $|G|=n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$.

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$ an abelian group of order $n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$. I have serious difficulties with ...
4
votes
2answers
187 views

Frattini subgroup of a finite group

I have been looking for information about Frattini subgroup of a finite group. Almost all the books dealing with this topic discuss this subgroup for p-groups. I am actually willing to discuss the ...
1
vote
1answer
324 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...