Tagged Questions

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
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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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$G$ 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 ...
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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 $n$, ...
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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 ...
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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 ...
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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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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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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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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 ...
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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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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 ...
1answer
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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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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 $A$-...
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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 ...
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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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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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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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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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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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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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is the given structure a group

is the below given structure a group?? [ * e a b c d f g ...
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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 ...
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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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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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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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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 $|M|=p^{n-1 }$ ?
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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. ...
1answer
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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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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 ...
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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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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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