A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold]

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1
8
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1answer
172 views

Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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2answers
34 views

$G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
0
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1answer
29 views

Linear representations of projective groups

Does the projective linear group $PSL_2(\mathbb{R})$ admit faithful linear representations? In other words, does there there exist a homomorphism $SL_2(\mathbb{R}) \to GL_n(\mathbb{R}),$ for some $n$, ...
0
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2answers
49 views

Non-abelian Order of $6$ is isomorphic to $S_3$ [on hold]

I know that it's duplicate, but , How can I prove it? I know that must be element "$a$" of order 2, and element "$b$" of order 3. What is the next step? In fact, from what I search it is claimed that ...
1
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1answer
9 views

number of orbits by action of $D_{12}$ on $\mathbb{Z}_{12}^k$

Let $X=\mathbb{Z}_{12}^k$ for $k\in \mathbb{N}$ and $G=D_{12}$. Define an action of $D_{12}$ on $X$ by setting rotations $r^n(p)=(p_1+n,\dotsc,p_k+n)$ where the coordinates are taken modulo $12$ and ...
3
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1answer
59 views

How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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0answers
28 views

Is power-associativity an equational property?

A magma $M$ is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as $x^mx^n=x^{m+n}$ for all $m,n$ positive integers and $x\in M$, ...
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2answers
38 views

Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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2answers
28 views

Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
3
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1answer
30 views

Problem with proof of $H \cap K $ is of finite index if $ H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...
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0answers
40 views

Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
1
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1answer
21 views

Conjugation of a direct product

Suppose that $H \times K \leq G = A\times B$ and let $g\in G$. Is it true that $g( H\times K)g^{-1} = gHg^{-1} \times gKg^{-1}$ holds in general?
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1answer
31 views

The sum of a family of submodules

I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions. Firstly, suppose $M$ is an $A$-...
0
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1answer
48 views

Kernel of homomorphism is normal subgroup

When trying to prove that a normal subgroup is a kernel, you map x to Nx(the coset). f(x) = Nx Why exactly do we do that? Why when you take an element in the normal ...
1
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1answer
49 views

Elements that are their own inverses in a symmetric group.

How many elements are their own inverses in $S_6$? I'm having a hard time figuring out how to calculate such a thing.
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1answer
38 views

Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
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2answers
53 views

Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
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3answers
38 views

Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
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0answers
25 views

Is this set a subgroup

I was reading lemma 3 of this proof. Is $P^p$ a subgroup ? We know that for any abelian group $G$, the set $G^n$ is a subgroup. But is it true for $P^p$ ?
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1answer
50 views

Not Abelian group G with Z(G) that contains only two elements? [closed]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
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4answers
51 views

is the given structure a group

is the below given structure a group?? [ * e a b c d f g ...
1
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0answers
41 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
2
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0answers
45 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
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2answers
31 views

Meaning of operation preservation in group homomorphism [closed]

What is meaning of operation preservation in a group homomorphism from a group $G_1$ to a group $G_2$?
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1answer
47 views

About the generators of a free group

Suppose we know that $G$ is a free group of rank $n$ and that $\{g_1,...,g_n\}$, with all the $g_i$ distinct, is a system of generators for $G$. Are we sure that it is also a $\textbf{free}$ system of ...
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0answers
28 views

cardinality of a maximal subgroup of a $p$ group

Let $P$ be a $p$ group with $|P|=p^n$. Let $M$ be a maximal subgroup of $P$. Is it true that $|M|=p^{n-1 }$ ?
2
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1answer
47 views

Kernel of a group action of rotation of a cube

Question: Let G be the rotation group of a cube Show that G has an action on a set of size 3. Well, if we consider axes through each opposite faces, then this set has only 3 possible axes. ...
1
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1answer
32 views

A sufficient condition for profinite groups

I know that Edwin Hewitt and Kenneth A. Ross (1970) show: Any compact Hausdorff torsion group is profinite. But I don't have the book, the proof seems long and I need only the case of abelian groups ...
1
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1answer
31 views

how to check if a subgroup is maximal?

Is there any general strategy to check whether a subgroup is maximal or not ? For example, in case of rings, we know that an ideal $I$ of a ring $R$ is maximal if and only if $R/I$ is a field. Is ...
4
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1answer
48 views

Prove that up-to isomorphism there are two integral domains of order $p^2$.

Prove that up-to isomorphism there is exactly one integral domain of order $p^2$ . Does there exist only two non-commutative rings of order $p^2$ upto isomorphism? We know that any group of ...
0
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1answer
36 views

Group Theory: Orders of Elements

Have been given following problem. Let G be group and x $\in G$. Prove that $x^2 = e$ if and only if x is of order one or two. My response. If order of x is 1, then by definition of order of an ...
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1answer
25 views

Prove all products of two disjoint 2-cycles are pairwise conjugate in $A_{n}$

Let $n \geq 4$. Prove that in the group $A_{n}$ of parity-even permutations of $\{1,2,...,n\}$, all products of two disjoint 2-cycles are pairwise conjugate. This is a past exam question for a ...
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2answers
66 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
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All 3-cycles are pairwise conjugate.

In my lectures on Group Theory our lecturer claimed the following: All 3-cycles are pairwise conjugate. He then went on to prove this but I am struggling with understanding his proof. I will try and ...
0
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3answers
47 views

Determining all the elements of S4?

What is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I tried ...
2
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1answer
48 views

how to calculate dimension of quotient space?

what is the dimension of $O(n) \setminus \mathbb{R}^n$ for $n \geq 2$? how to calculate this? $n - \frac{n(n-1)}{2}$ does not work well.
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0answers
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Writing in disjoint cycles

Write $\left ( 123 \right )^{-1}\left ( 23 \right )\left ( 123 \right )$ in disjoint cycles. I get that $\left ( 123 \right )^{-1}$=$\left ( 132 \right )$ $\left ( 132 \right )\left ( 23 \right )\...
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1answer
49 views

Group with Elements of Order 2 [closed]

How can I prove that if a group, all the elements are from the order of $2$, then is isomorphic to $Z_2+Z_2+Z_2+..+Z_2$.
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0answers
17 views

What is the point of $P'$ in this proof of Groups of order $pq$.

I refer to this site: http://planetmath.org/groupsoforderpq In case 2, it is stated that $\text{Aut}(Q)$ has a subgroup $P'$ of order $p$, where $P^{{\prime}}={\{x\mapsto x^{i}\mid i\in\mathbb{Z}/q\...
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0answers
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Is there only one internal semidirect group?

I am a bit confused on the concept of internal semidirect group $N\rtimes H$. For external semidirect group $N\rtimes_\varphi H$, I know that there can be possibly many semidirect groups depending on ...
2
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0answers
34 views

Deriving an Element of the Lorentz Group SO(1, 3)

We know that $SO(1, 3)$ is isomorphic to $SU(2) \otimes SU(2)$: $$SO(1, 3) \cong SU(2) \otimes SU(2)$$ We also know that $$ \left(\frac{1}{2}, \frac{1}{2}\right) = \left(\frac{1}{2}, 0\right) \...
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0answers
23 views

Actions of unipotent groups

If we have an connected unipotent algebraic group $G$ over $\mathbb{F}$ (the algebraic closure of a finite field of characteristic $p>0$), with an $\mathbb{F}_q$ structure (where $\mathbb{F}_q$ is ...
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intersection of two sets equals an LCM set

I'm interested in how to prove the following: For every $n\in \Bbb N$, let $B_n = \{n·k| k\in \Bbb N\}$ where $\Bbb N$ is the natural numbers group excluding $0$. Prove that $B_n∩B_m=B_{c(...
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How to write semidirect product as “presentation” notation

I am referring specifically to this example http://planetmath.org/groupsoforderpq In Case 2, the group should be $\mathbb{Z}_q\rtimes\mathbb{Z}_p$. How do we write it in the presentation of $G={\...
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0answers
14 views

Result on a subgroup of a direct product

Suppose that $G=A \times B$ is finite group as a direct product of $A$ and $B$ and let $U \leq G$ such that $A \subseteq U$. Then $U = A \times \pi_{B}(U)$ Clearly, $A\unlhd U$ and $A \cap \pi_{B}(U) ...
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1answer
20 views

When does prime $p$ divide a term in the numerator and denominator of ${p^m \cdot k}\choose {p^m}$ the same number of times?

This has questions comes to me via a proof to Sylow's First Theorem where $G$ is a finite group of order $p^m \cdot k$, the number $p$ is a prime divisor of $|G|$, and $p^m$ is the highest power of $p$...
2
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1answer
19 views

No subgroup of $S_n$ containing stabilizier of 1?

Is it true that the stabilizer of $1\in \left\{1,\dots ,n \right\}$ in $S_n$ is a maximal subgroup? Intuitively I'm thinking that as soon as you add another permutation, you'll somehow be able to ...
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What happens if we do $H\rtimes N$, the other way around for semidirect product?

The usual way for semidirect product is $N\rtimes H$, where $N$ is a normal subgroup of $G$, $H$ is a subgroup of $G$. What happens if we try $H\rtimes N$? Where $H$ is not a normal subgroup of $G$.(...
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24 views

Result on direct product of groups

Suppose that $G=A \times B$ is finite group as a direct product of $A$ and $B$ and let $U \leq G$. If $N \leq U$ for all $N\unlhd A$ then $A\subseteq U$. I start by letting $a\in A$ be arbitrary. If $...