0
votes
0answers
12 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 ...
-1
votes
0answers
15 views

How to show that there are as many left cosets as there are right cosets?

G is a finite group and H is a subgroup, How to show that there are as many left cosets of H as there are right cosets?
0
votes
0answers
37 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 ...
2
votes
0answers
39 views

Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ such that $|S\cap T|$ is as larger as possibe and write $D = S \cap ...
0
votes
1answer
25 views

Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
1
vote
2answers
33 views

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$.

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$. Also another question, do p and q have to be distinct for this to hold? On top of ...
0
votes
2answers
35 views

Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...
7
votes
1answer
89 views

Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$.

Question : Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
0
votes
1answer
31 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
1
vote
1answer
32 views

Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$

Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$ Attempt: $Z_p \oplus Z_p$ has $p^2-1$ elements of order $p$ . Hence, all non trivial elements of $Z_p \oplus ...
2
votes
2answers
25 views

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
1
vote
1answer
38 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
1
vote
1answer
31 views

Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
1
vote
1answer
38 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
2
votes
1answer
42 views

Visualizing the 48 actions of GL(2,3)

Hello and thank you for your patience. (DISCLAIMER: I'm a novice and not a mathematician by trade and I'm not certain how to articulate most of my questions here. I am learning from experiences and ...
2
votes
2answers
53 views

What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
1
vote
1answer
38 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
3
votes
1answer
47 views

How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
2
votes
1answer
60 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
2
votes
0answers
50 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
0
votes
0answers
27 views

To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
1
vote
1answer
50 views

Is this type of a subgroup always normal? [duplicate]

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...
0
votes
1answer
51 views

If G is a finite group and $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$

Let G be a finite group. Show that, given $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$. I'm trying to use the info that is finite, but I can't find a way. For instance, if G is a ...
3
votes
2answers
79 views

What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
2
votes
2answers
71 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
0
votes
1answer
21 views

Questions on proof that transvections are conjugate in $GL(V)$.

I have difficulty following the proof that transvections are conjugates in $GL(V)$, and for $n \ge 3$ even in $SL(V)$. I give the necessary definitions and the proof, with the problematic parts ...
2
votes
0answers
38 views

Groups with single conjugacy class of subgroups. [duplicate]

I wish to know all those groups in which there is single conjugacy class of subgroups of fixed order.For example, In finite cyclic groups and in Alternating group of degree 4, number of conjugacy ...
0
votes
1answer
37 views

Find all sub groups of order $4$ in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$ . Are they all cyclic?

Find all sub groups of order 4 in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$. Solution : $\mathbf{Z}_4 =\{0,1,2,3\}$ $O(1) = O(3) = 4$, $O(0) = 1$, $O(2) = 2$ Hence, I found the subgroups of order 4 as ...
-2
votes
0answers
32 views

$|G| \geq 3 $ is finite and has subgroup with index more than 4 then $G$ is not simple.

suppose $G$ is a finite group with more than three elements,prove that if $H$ will be a proper subgroup of $G$ with this property that $[G:H] \geq 4$ then $G$ is not simple. I found this question a ...
1
vote
1answer
63 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
0
votes
1answer
23 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
2
votes
1answer
34 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
0
votes
1answer
30 views

$p$-subgroups conjugate iff $\cong$ to Sylow p-subgroups of some other groups?

Let $G$ be a finite group and $p$ a prime such that $p^\alpha$ divides $|G|$ and $p^{\alpha+1} \nmid |G|$. I know that Sylow $p$-subgroups of $G$ are conjugate to one another but if we have some ...
1
vote
3answers
69 views

Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am lookking for ...
0
votes
0answers
34 views

Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
0
votes
1answer
51 views

Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
1
vote
4answers
46 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
1
vote
2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
2
votes
2answers
79 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
4
votes
1answer
121 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
0
votes
0answers
22 views

Sylow subgroups and normalizers [duplicate]

Let $H$ be a Sylow 3-subgroup and $K$ a Sylow 5-subgroup of a finite group $G$. Suppose $|H|=3$ and $|K|=5$ and $N_G(K)$, has an element of order 3. Show that $N_G(H)$ has an element of order 5.
0
votes
1answer
37 views

Is this object a group?

$\begin{array}{ccccccccc} \times&e_0&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ e_0&I&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ ...
0
votes
1answer
15 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
1
vote
0answers
21 views

Let $G$ be a group of order 24 and suppose $n_2(G) > 1 \ \ and \ \ n_3(G) > 1$ . Then $G \cong S_4$

My attempt is : Since $n_3 > 1$ and $n_3 \equiv 1 \ \ mod \ \ 3 $ and divides 8, then the only possibilty is $n_3 = 4$ and thus $| G:N| = 4$, where $N = N_G(P)$ and $P \in Syl_3(G)$. Then $G/K $ ...
4
votes
2answers
103 views

Why free presentations?

What is the motivation to study "free" presentations of groups,even though all (or almost all) the questions (or the problems) concerning this type of presentations are known to be undecidable ?
1
vote
2answers
74 views

$g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
0
votes
1answer
50 views

Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
3
votes
1answer
32 views

Having trouble grasping the class equation as an explanation as to why a conjugate class's order divides the order of a group.

Suppose $|G|$ is a prime power $p^n$ and that $N$ is a normal subgroup of $G$. Show that $|y^G|$ is a power of $p$ whenever $y \in G$ Attempt: Firstly, I assume that $y^G = \{ gyg^{-1} | g \in G ...
3
votes
1answer
41 views

Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. ...
1
vote
1answer
31 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...