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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Every subnormal of a semisimple is normal

There is a famous lemma saying: "Every subnormal subgroup of a semisimple group is normal and is a semisimple direct factor of the group". Any hints about it? :)
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Subgroups Lattice of Automorphism Group of Linear Groups

Could someone tell me the subgroups lattice of $PSL(n,p^k)$ or could someone tell me sources I should read to know all about $Aut(PSL(n,p^k))$ and its subgroups lattice. Thanks in advance.
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131 views

Representation theory question + SU(n)

Would you please help me in how to solve these questions : A- Let $H$ be a subgroup of a finite group $G$.Let $\alpha$ and $\beta$ be class function of $G$ and $H$ respectively. Prove that ...
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0answers
186 views

A question on a generalization of perfect numbers

First of all, I would like to call a group immaculate provided that the orders of $G$ and $\Sigma$ (the order of $N$) where $N$ varies over all normal subgroups of $G$, are equal. From here it has ...
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0answers
218 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
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1answer
60 views

Some questions about Banach Tarski proof

Banach-Tarski proof as been the topic of a video by the well-known Youtube channel VSauce but there were some parts that I didn't understand. So I went reading for the proof on Wikipedia, and I didn't ...
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1answer
42 views

Help proving inequality

3. $\delta$ is the standard Euclidean valuation on $\mathbb{Z}[i]$. For each of the following pairs $a,b \in \mathbb{Z}[i]$, find $q,r \in \mathbb{Z}[i]$ such that $a = qb + r$, where either $r = ...
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2answers
73 views

What can we say about the order of a group?

Let $G$ be a group and $a ∈ G$. If $a^{12}= e$, what can we say about the order of $a$? What can we say about the order of $G$? We know that $|a|$ divides $12$, but what can we say about the order of ...
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3answers
129 views

abstract Algebra (group theory)

Three coins are placed on a table; showing heads. Can you get all the coins to show tails, by turning over two coins at a time? Use Group Theory to prove your answer. I know that the answer is no I ...
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2answers
49 views

Can these two quotient groups be isomorphic?

Let $N$ and $M$ be two normal subgroups of a group $G$. Then we can show that the set $NM \colon= \{\ nm \ | \ n \in N, \ m \in M \ \}$ is a subgroup of $G$, that $M$ is normal in $NM$, and that $N ...
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3answers
324 views

Example of Intersection of Pure Subgroup which is not Pure

i have learn that intersection of pure subgroup of a group G is not necessarily pure. Can someone show me an example when such a case exists? I'm aware that if G is torsion-free, then intersection of ...
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3answers
89 views

What does “nilpotent” in a “nilpotent group” mean?

It seems to have nothing to do with the usual nilpotency, i.e. $\exists n\in \mathbb{N}:x^n=0$. Actually I think the latter only makes sense in a ring or more rich structure. I tried to relate some ...
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3answers
204 views

$GL_n(\mathbb{R})$ is cyclic?

Is the set $GL_n(\mathbb{R})$ is cyclic group? It is not, right? $GL_1(\mathbb{R})$ is just $\mathbb{R}^{*}$ which is dense. I forget, but there was maybe a result which said something like any ...
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8answers
524 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
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6answers
270 views

The product of finitely many cyclic groups is cyclic

How to prove that the direct product of finitely many cyclic groups $C_{n_1}\times C_{n_2}\times\cdots\times C_{n_m}$ is cyclic if the $n_i$'s are pairwise relatively prime?
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4answers
208 views

Prove that if $a$ and $b$ are distinct elements of a group $G$, then $a^2 \ne b^2$ or $a^3 \ne b^3$.

Prove that if $a$ and $b$ are distinct elements of a group $G$, then $a^2 \ne b^2$ or $a^3 \ne b^3$. Im not really sure how to approach this, any thoughts? Thanks
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4answers
365 views

Can an empty set be both torsion and torsion free group?

I was wondering if an empty set can be a torsion group (since the definition of torsion group is that if $x$ is in the set $X$ has a finite order. However, the assumption is false, so the implication ...
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4answers
336 views

2 Tricks to prove Every group with an identity and x*x = identity is Abelian - Fraleigh p. 48 4.32

I didn't see the similar question. My question isn't a duplicate because I understand how to prove this. But I don't understand where the hint comes from. That question doesn't flesh it out. ...
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3answers
1k views

How to prove that $A_5$ has no subgroup of order 30?

I need to show that $A_5$ has no subgroup of order 30. Any ideas?
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5answers
144 views

Is it true, $O(ab)=O(ba),$ Where $G$ is a group and $a,b \in G.$

Suppose $O(a)$ and $O(b)$ is finite and also $O(ab)$ and $O(ba)$ is finite. Then L.C.M $(|a|,|b|)= L.C.M (|b|,|a|).$ (Is that Correct ?) Suppose $O(a)$ and $O(b)$ is finite but $O(ab)$ is infinite, ...
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4answers
1k views

Why is the set of natural numbers considered a non-Abelian group?

I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.
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6answers
3k views

Prove that: set $\{1, 2, 3, …, n - 1\}$ is group under multiplication modulo $n$?

Prove that: The set $\{1, 2, 3, ..., n - 1\}$ is group under multiplication modulo $n$ if and only if $n$ is a prime number. (Do not use Euler phi function)
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6answers
483 views

Let G be a group and a; b ∈ G. Suppose |a| = |b| = |ab| = 2. Then show that ab = ba.

I'm having trouble understanding this question and help would be appreciated. If |ab|=2 and |a|=2, |b|=2, wouldn't this imply that |a||b|=|ab|=4? How would I go about proving that this is Abelian? ...
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4answers
130 views

If all Subgroups are Cyclic, is group Cylic? [duplicate]

I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic.
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5answers
155 views

Why does $1 \cdot 0=0$ not stand?

A set $G$ together with an operation $*$ is called group when it satisfies the following properties: $a*(b*c)=(a*b)*c, \forall a,b,c \in G$ $ \exists e \in G: e*a=a*e=a, \forall a \in G$ $\forall ...
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4answers
104 views

Why is $\langle \mathbb{Z}_4, + \rangle$ not isomorphic to $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$?

I'm having some trouble here, specifically with the idea of $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$ as a group. Can anyone help me out with some explanations? Moreover, I generally ...
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3answers
333 views

Irrationals: A Group?

I understand that the set of irrational numbers with multiplication does not form a group (clearly, $\sqrt{2}\sqrt{2}=2$, so the set is not closed). But is there a proof or a counter-example that the ...
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5answers
198 views

Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.

Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not. pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times ...
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3answers
97 views

Prove that $\langle a \rangle = \langle a^{-1} \rangle$

Let $G$ be a group and let $a \in G$. Prove that $\langle a \rangle = \langle a^{-1} \rangle$. (where $\langle a \rangle = \{\ldots,-2,-1,0,1,2,\ldots\} = \{a^k : k \in \mathbb{Z}\}$) ...
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226 views

Why $(\mathbb{Z}, + )$ is not isomorphic to $(\mathbb{R}^+, *)$?

Can someone explain to me why $(\mathbb{Z}, + )$ is not isomorphic to $(\mathbb{R}^+, *)$ where $*$ is multiplication. My book says they aren't really isomorphic and doesn't say why. I thought that ...
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4answers
455 views

Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.

This solution is from here and yahoo. Given $a,b$ elements of $G$, and $ab$ has finite order $n$. Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$. Need to show $n$ is the smallest positive integer ...
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2answers
314 views

Find number of abelian groups of order $27$?

I was trying to solve the following problem: Find number of abelian groups of order $27$ ? Could someone point me in the right direction? Thanks in advance for your time.
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154 views

Semi-direct product of different groups make a same group?

We can prove that both of: $S_3=\mathbb Z_3\rtimes\mathbb Z_2$ and $\mathbb Z_6=\mathbb Z_3\rtimes\mathbb Z_2$ So two different groups (and not isomorphic in examples above) can be described as ...
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3answers
244 views

Find all possible abelian groups of order $120$.

Find all possible abelian groups of order $120$. If someone could walk me through how to do this, that would be great.
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3answers
255 views

In a finite abelian group of order n, must there be an element of order n?

In a finite abelian group of order n, must there be an element of order n? This question is bugging me, I've been thinking about it all afternoon long. Can somebody hint me? I guess it is true, but ...
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4answers
668 views

If the size of 2 subgroups of G are coprime then why is their intersection is trivial?

Let H and K be subgroups of G, with size p and q respectively, where p and q are coprime, how can we show that H intersect K is {e} where e is the identity element in G
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4answers
763 views

Vector space over $\mathbb{Q}$ or $\mathbb{Z}$?

I am looking at the following: Show that a torsion-free divisible group $G$ is a vector space over $\mathbb{Q}$. I have no problem verifying the axioms of vector spaces after noting that the ...
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4answers
64 views

If $G/H$ is a group, then does $H$ have to to be normal?

If $H$ is normal then $G/H$ is a group. But is the converse true? That is, if $G/H$ is a group, does this mean that $H$ is normal? Or are there any counterexamples?
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4answers
94 views

Finding all groups of order $7$ up to isomorphism?

I'm learning group theory but I didn't learn any concepts of building groups. I know that there exists the identity group $\{e\}$, and the group with 2 elements: $\{e,a\}$. If I try to create a ...
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6answers
265 views

Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
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3answers
1k views

A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$

How to show that: A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$.
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4answers
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Proving that a set of matrices is an abelian group

Prove that the set of matrices in the form of $\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha &\cos \alpha \end{array}\right]$ (while $\alpha \in R$) with the ...
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3answers
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Show that all abelian groups of order 21 and 35 are cyclic.

Show that all abelian groups of order 21 and 35 are cyclic. I have no idea on how to start. Can anyone give some hints?
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3answers
1k views

The Direct Product of a finite number of cyclic groups, is again a cyclic group?

Can anyone please brief me on this question, do you prove or disprove? If it is to be disproved can you give a counterexample? if there is no generator, then what is an example of such a Direct ...
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2answers
1k views

If every cyclic subgroup of $G$ is normal so is every subgroup?

I searched the Web to find an conterexample for this statement: If every cyclic subgroup of group $G$ is normal in $G$ then every subgroup of $G$ is normal in $G$. But couln't find any. It seems it ...
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3answers
330 views

Proving $H$ is normal in $G$. [duplicate]

Possible Duplicate: Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$. I have to solve the following problem. It's an ...
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2answers
279 views

Epimorphism from GL(2,Z) to GL(2,Z)

Is there an epimorphism $f\colon \mathrm{GL}(2,\mathbb{Z})\to \mathrm{GL}(2,\mathbb{Z})$ which is not injective? Here, $\mathrm{GL}(2,\mathbb{Z})$ is the group of invertible $2\times 2$ matrices with ...
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2answers
365 views

What is the center of $V$, the Klein 4 group?

Please help - in my notes, it is the group $V$ itself. I just want to confirm this. Can you also explain and give an example if that is possible?
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4answers
68 views

In a group $G$, $a$ is the only element of order $n$, for some $n\in \mathbb N$. Prove that $a\in Z(G)$.

If In a group $G$, the element $a\in G$ is the only element of order $n$, i.e., $a^n=e$ for some positive integer $n$. Then we have to show that $a\in Z(G)=\{x\in G : xg=gx, \forall g\in G\}$. How ...
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2answers
113 views

If $g\neq g^{-1}$ for all $g\in G\setminus \{e\}$, then the order of $G$ is odd

Given a group $G$ where $g\neq g^{-1}$ for all $g$ other than the identity, show the order of $G$ is odd. What does it mean for the order of group $G$ to be odd? Any help welcome.