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.

learn more… | top users | synonyms (2)

-4
votes
0answers
29 views

Number of elements of order $2$ in a group of even order

If $G$ is a group of even order, then the number of elements of order $2$ in $G$ is: (a) 2 (b) 4 (c) even (d) odd
-2
votes
0answers
15 views

Explanation of definition of normal subgroup

Recently I am studying group theory to know about Orbit-Stabilizer Theorem and I'm a novice learner.I got a definition of normal subgroup as mentioned in the link: ...
2
votes
3answers
21 views

Extend isomorphism of subgroups to homomorphism of groups

Given two finite groups $G_1$ ang $G_2$ with respective subgroups $H_1$ and $H_2$ satisfying $H_1\cong H_2$ via the isomorphism $\phi$, is it always possible to extend $\phi$ to an homomorphism ...
3
votes
2answers
47 views

Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, ...
1
vote
0answers
18 views

For any subnormal subgroup of a finite group, must its normalizer be subnormal, too?

Let $G$ be a finite group and $H$ a subnormal subgroup of $G$. Must $N_G(H)$ be subnormal, too?
2
votes
1answer
29 views

Associative Proof

I have a non-empty set $R$ with a binary operation $*$. If the pair has an identity element $e\in R$ and $$(a*b)*(c*d)=(a*c)*(b*d)$$ holds for all $a,b,c,d \in R$, how do I then prove that this is ...
1
vote
2answers
27 views

Identity Element and Identity Properties [duplicate]

Learning more abstract algebra, really not the most enjoyable of subjects, as nothing seems all that clear cut, but here goes anyway. I have a set $\mathbb Q = \{{p \over q} : p,q\in \mathbb Z \text{ ...
0
votes
3answers
64 views

Is $\ker(\operatorname{nat}_H)=H$?

This question came in the exam today, sadly I couldn't answer it. The question said: Prove whether or not this is a true statement, stating the reason. $$\ker(\operatorname{nat}_H)=H$$ where ...
1
vote
0answers
27 views

Subgroup of group is normal [duplicate]

This question came in the exam today, unfortunately I couldn't answer it. The question said: Proof whether or not this is a true statement, stating the reason. Subgroup of group is normal I ...
1
vote
1answer
21 views

Distributing infinite supply of $n$ distinct objects into $k$ identical urns

I have $n$ distinct objects, namely {$n_{1\le i \le n}$} with an infinite supply of each of them, and I have $k$ identical, indistinguishable urns to place the objects in. Each urn will contain ...
1
vote
1answer
36 views

In $S_{3}$ what is the group generated by $(123)$?

In $S_{3}$ what is the group generated by $(123)$? Is there a way to find the elements of the group generated by $(123)$?
1
vote
0answers
35 views

Question about direct product of two groups.

Let $G=\mathbb{Z}_n \times \mathbb{Z}_m$ and $d=p^k$ for some prime $p$ such that $d$ divides both $n$ and $m$. Then $G$ has exactly $d\phi(d)+[d-\phi(d)]\phi(d)$. For example consider the group ...
0
votes
1answer
19 views

On special cyclic subgroups of a finite group

Let $G$ be finite group and $x,y\in G$. We know that $\langle x, g\rangle$ is cyclic for any $g\in G$. Also we have $\langle y, t\rangle$ is cyclic for some $t\in G$. Now prove that $\langle xy, ...
0
votes
2answers
27 views

Proof involving Cyclic group, generator and GCD

Theorem: $$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$ Let G be a group and $$ a \in G$$ such that $$|a|=n$$ Then: $$\left\langle a^k \right\rangle = \left\langle ...
8
votes
2answers
183 views

Divisor of a finite group

Suppose we have a finite group $G$ and $d\in \mathbb N$ is a divisor of $|G|$. We define the set $E_d= \{g\in G : g^d =1\}$. Prove that $d$ is also a divisor of $|E_d|$. So far I proved that ...
3
votes
1answer
28 views

The relationship of subnormal subgroups and modular subgroups of a finite group.

Let $G$ be a finite group, a subgroup $H$ of $G$ is called subnormal if it's a term of a composition series of $G$, and is called modular if it's a modular element of the subgroup lattice $L(G)$. My ...
1
vote
1answer
32 views

Groups occuring as derived subgroups.

I want to prove this problem but I have no idea how to start it. If you know please hint me, thanks. Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the ...
6
votes
2answers
65 views

Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
2
votes
2answers
39 views

Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group Unlike this question: To show that the center is a ...
0
votes
0answers
14 views

How is the first Sylow theorem a strenghtening of Cauchy's theorem? [duplicate]

Taken from Wikipedia: Theorem 1: For any prime factor p with multiplicity $n$ of the order of a finite group $G$, there exists a Sylow $p$-subgroup of $G$, of order $p^n$. The following ...
1
vote
1answer
16 views

is “being reductive” extension-closed?

Suppose we have a short exact sequence of linear algebraic groups over a field of characteristic zero $$1 \to N \to G \to G/N \to 1$$ with $N$ and $G/N$ reductive (that is connected with trivial ...
0
votes
1answer
34 views

Criterion for $a^i=a^j$ proof

Let G be a group and let a be an element in G. If a has infinite order, then $$a^i=a^j$$ if and only if $i=j$ If $a$ has finite order, say $n$, then $$\left \langle a \right ...
0
votes
6answers
34 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
2
votes
1answer
53 views

What is the use and motivation for this particular concept in permutations?

Say you have the permutation $(54231)$ element of $S_5$ Now you drop say the "4" and then re-rank the remnant permutation on the other elements. Then you are left with, $(4231)$ element of $S_4$ ...
0
votes
0answers
29 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
4
votes
1answer
79 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
5
votes
2answers
93 views

How many ways to select $k$ vertices of an $n$-gon?

I have a regular $n$-gon, of which I have to select $k$ vertices. The selections must be rotationally distinct; two selections would be considered equivalent if one is a rotation of the other. For ...
2
votes
1answer
23 views

An $RO$-group which is not $O$-group

I was thinking of some example for an Right ordered group ( $RO$-group) which is not an $O-$group (Ordered group) i.e. not left ordered. I guess looking in matrix groups will be fruitful but how to ...
4
votes
1answer
52 views

Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?

Is there an example in which $G$ is a finite abelian group and $\textrm{Aut}(G)$ is abelian but $G$ not cyclic?
2
votes
2answers
43 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it. [on hold]

Problem Let G be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
2
votes
3answers
55 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
1
vote
1answer
32 views

Natural action of $S_n$ on $\{ 1,2,\dots,n \}$

From reading online the "natural" action of $S_n$ on $\{ 1,2,\dots,n \}$ is $(g,x) \mapsto gx$. How is this action transitive? As far as I can see if we take $g$ to fix some element we will not get a ...
1
vote
1answer
40 views

Coproduct of groups explanation

Could someone please explain the following? "Let $G=\prod G_{i}$ be a direct product of groups. Then each $G_j$ admits an injective homomorphism into the product, on the j-th component, namely the map ...
-1
votes
1answer
55 views

Is $O(n)$ normal in $GL(n)$?

Is the orthogonal group $O(n)$ normal in $GL(n)$? Here is what I did so far: Let $Q\in O(n),S\in GL(n)$ we want to check if $S^{-1}QS\in O(n)$: $(S^{-1}QS)^T=(S^{-1}QS)^{-1}\iff ...
2
votes
2answers
36 views

Is this a correct way to think about specific examples of groups using the category theory definition?

I'll say now, before anything else, that I probably don't know what I'm talking about. This is more me making a (hopefully) educated guess about a topic I'm not too familiar with. I recently started ...
-1
votes
1answer
27 views

Center of group of a dihedral group

An example from my text ask to verify this: $$Z(D_{n})= \begin{cases} {R_{0},{R_{180}}} & \text{when n is even}\\ {R_{0}} & \text{when n is odd}\end{cases}.$$ How should I begin to verify ...
1
vote
1answer
35 views

Proof that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$

I am trying to prove that Prove that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$ Unlike this question: Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are ...
2
votes
2answers
34 views

Can every torsion-free nilpotent group be ordered?

I know that a torsion-free abelian group can be ordered and have done two proofs for that too. But the next two question that popped up in my mind were- Can every torsion-free nilpotent group be ...
0
votes
1answer
50 views

What is orthogonal group $O(1)$?

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?
3
votes
2answers
66 views

Is the left translation $T_a(x) =ax $ a homomorphism?

I apologize if this is a super basic question but I was reading Lang's undergraduate algebra book and it says that the following function is a homomorphism: $$T_a(x) = ax$$ The way I would check if ...
1
vote
0answers
35 views

If $G$ and $H$ are groups, prove that $(G \times H, x)$ is a group.

Prove that, if $(G,\ast)$ and $(H,\bullet)$ are groups, then the Cartesian Product $G \times H$ with the operation $(g_1,h_1) \circ (g_2, h_2) := (g_1 \ast g_2, h_1 \bullet h_2)$ $(G \times H, ...
0
votes
1answer
13 views

Order of Hom$(D_n,\mathbb{C}^*)$

What is the order of Hom$(D_n,\mathbb{C}^*)$? I know that $D_n/[D_n,D_n]$ is isomorphic with $\{\pm 1\}$ if n is odd and isomorphic with $V_4$ if n is even. And I know that $\#D_n=\#D_n/[D_n,D_n]$. ...
0
votes
1answer
42 views

Specific topological example of Nielsen-Schreier theorem

I'm assuming that the following question should be basically trivial, and that I'm just misunderstanding something basic, but some clarification would be much appreciated. There is a section in my ...
2
votes
2answers
41 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
1
vote
0answers
41 views

Reference Request : Quotients of nilpotent groups which are torsion free

I am currently writing my thesis and looking for a reference (or a short proof) to the following fact: Let $N$ be a finitely generated nilpotent group, and denote its central series by ...
1
vote
1answer
48 views

Is there a term describing an almost-group without closure?

(Background: I'm working on a dissertation in music theory that involves group theory in a small way, but it's mostly out of my element.) I'm working with the group $\mathbb{Z}_8$, but I'm wondering ...
1
vote
2answers
14 views

Proof of the right and left cancellation laws for Groups

I was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first ...
1
vote
1answer
31 views

How can you tell if a normal subgroup induces a semidirect product?

Suppose I have some (finite) group $G$ and a normal subgroup $N$. I know there's no full characterization of whether $G \cong N \rtimes G/N$, but are there well-known tests I can use to answer the ...
2
votes
1answer
21 views

If a finite $p-$ group has only one composition series show that it is cyclic

If a finite $p-$ group has only one composition series show that it is cyclic. What I tried: Let $G$ be a finite p-group .Then $|G|=p^n$ for some $n$ .Now every group of order $p^{n-1}$ is normal in ...
2
votes
1answer
23 views

Proof that $N=\langle \{g^2\}\rangle$ normal

How do I prove that for a group $G$ and a subgroup $N$ that is generated by $S=\{g^2: g\in G\}$, that $N$ is normal? Also: how do I know then that $G/N$ is abelian?