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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$a^2 + b^2$ never leaves remainder $3$ when divided by $4$

Already did something like that to prove the square of an even number Always leaves remainder $1$ when divided by $8$, in which I used induction to arrive at the result. However, I don't know how to ...
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19 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 ...
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1answer
11 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 ...
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1answer
34 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 $(123)$?
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30 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 ...
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1answer
18 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, ...
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26 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 ...
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164 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 ...
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23 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 ...
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1answer
30 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 ...
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63 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 ...
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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 ...
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13 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 ...
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1answer
15 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 ...
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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 ...
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6answers
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$\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 ...
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1answer
48 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 $S_4$ ...
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28 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} ...
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76 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?
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86 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 ...
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1answer
20 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 ...
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51 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?
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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). ...
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3answers
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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 ...
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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 ...
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1answer
39 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
29 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 ...
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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 $O(1)$?
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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 ...
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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, ...
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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 $\#D_n=\#D_n/[D_n,D_n]$. ...
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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 ...
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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 ...
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38 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...