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)

3
votes
1answer
30 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
22 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 For the identity, $e$ clearly is in $Z(G)$ and in ...
0
votes
0answers
12 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
0answers
7 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
33 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
28 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
0answers
31 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
15 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} ...
3
votes
1answer
46 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?
2
votes
1answer
19 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
49 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
35 views

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

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
51 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
29 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 ...
0
votes
1answer
36 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
53 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 ...
1
vote
2answers
34 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
26 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
28 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
65 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
34 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
41 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
1answer
33 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
29 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
22 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?
-1
votes
2answers
34 views

Is the subgroup of a non-abelian group is non-abelian?

Is the following statement always true Subgroup of a non-abelian group is non-abelian
3
votes
2answers
52 views

Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
4
votes
1answer
47 views

Classifying groups of order 18

I am trying to classify groups of order 18. So far, I have shown that a group $G$ of order 18 is given by $G\cong C_9 \rtimes_{\varphi} C_2$ or $G\cong (C_3 \times C_3)\rtimes_{\varphi} C_2$. If ...
0
votes
0answers
58 views

Showing conjugacy between the Tent Map and another

" Show that the tent map: xn+1 = 1 - |1 - 2xn| is a conjugate to the map defined as follows: θn+1 = Nnθ0 mod1 also: xn = sin2[π*θn] and θ0 = π-1arcsin[x01/2] " I'm really struggling to show ...
1
vote
0answers
6 views

How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
7
votes
1answer
68 views

Groups with finite automorphism groups.

An easy argument shows that for any finite group $G$ the cardinal of $Aut(G)$ is less than $(|G|-1)!$. In particular the automorphisms group of a finite group is finite. Basically my question is about ...
2
votes
2answers
49 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...
2
votes
1answer
23 views

Proof that there exists an $x \in G$ such that $xa = b$

So this is my challenge: Let $G$ be a group and $a,b \in G$. Then $xa = b$ has a unique solution I went about saying that $xa = b \iff xaa^{-1} = ba^{-1} \iff x = ba^{-1}$. $ba^{-1} \in G$ ...
0
votes
3answers
105 views

What is the intuition behind the definition of the kernel of a homomorphism

I was starting to study some algebra (groups and homomorphisms in particular) and came across the definition of the kernel (for a group-homomorphism $f:G \rightarrow G'$): $$\ker(f) = \{ x \in G \mid ...
-1
votes
1answer
104 views

What is the mathematical difference between group and category?

This question is quite similar to the following link: Why learn Category Theory in order to study Group Theory? The above link is nice but I could not find the difference mathematically between ...
2
votes
1answer
32 views

Is there any cyclic subgroup of order 6 in in $ S_6$?

Is there any cyclic subgroup of order 6 in $ S_6$? Attempt: $|S_6|=6!=720$ Let $H$ be a subgroup of $S_6$ ,$H$ cyclic $\iff\langle H \rangle=\{e,h,h^2,...,h^{n-1}\}=S_6$
1
vote
1answer
49 views

Find the Subgroup of $\mathbb Z_4 \times \mathbb Z_2$ (Joseph A. Gallian - Exercise - 8.22)

Find the Subgroup of $\mathbb Z_4 \times \mathbb Z_2$ that is not the form of $ H \times K$, where $H$ is a subgroup of $\mathbb Z_4$ and $ K$ is a subgroup of $\mathbb Z_2$ Order ...
2
votes
2answers
60 views

Finite groups whose non-trivial elements have no fixed points

(I first asked this question on MathOverflow, but was recommended to ask here at Mathstackexchange instead.) I am interested in finite groups $G$ acting on a finite set $X$ with the following ...
0
votes
0answers
22 views

Finite groups whose non-trivial elements have no fixed points [duplicate]

I am interested in finite groups $G$ acting on a finite set $X$ with the following property: (*) fix(g)=$\emptyset$ for all $g\in G\setminus\{1\}$, where fix(g):=$\{x\in X|gx=x\}$ denotes the set ...
2
votes
2answers
32 views

$H,N(H)$ are subgroups of $G$ show that $H\lhd N(H)$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ $N(H)$ is also subgroup of $G$. I need to prove that $H$ is a normal subrgoup in $N(H)$ Attempt: $H\lhd N(H) \iff ...
0
votes
1answer
46 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
-1
votes
2answers
42 views

Proving each automorphism of a group $G$ fixes a normal subgroup of order $p^n$ if $p\nmid\frac{|G|}{p^n}$

I have been going through Herstein's Algebra and came across this problem: "$G$ has order $p^{n}m$ where $p$ is a prime, $p$ doesn't divide $m$. Suppose $G$ has a normal subgroup $P$ of order $p^n$. ...
1
vote
1answer
33 views

'Large' closed subgroup

I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold $(M,h)$ admits a large closed subgroup $K$ of the ...