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)

1
vote
0answers
7 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 $N_r$, that ...
1
vote
1answer
18 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
12 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
16 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
14 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
19 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
30 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
1answer
28 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
29 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
11 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
5 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
63 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
45 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
22 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
100 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
97 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
29 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
48 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
54 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
20 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
43 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
40 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
32 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 ...
2
votes
1answer
28 views

The abelian group of smallest order and smallest non prime integer n divides |G| but G doesn't have an element of order n?

I don't know how to think of an example. What's an example of such a group. It doesn't make sense to me because if it is a finite abelian group, it can be written as a direct product of the integers ...
2
votes
3answers
53 views

How is $\lbrace a_1, a_2, …, a_n : a_i \in \Bbb Z_2\rbrace$ a group?

I was asked to prove that if we define \begin{equation*} \Bbb Z_2^n = \lbrace a_1, a_2, ..., a_n : a_i \in \Bbb Z_2\rbrace \end{equation*} then it's a group under the operation of addition like ...
3
votes
2answers
28 views

Show that $N(H):=\{g\in G; gHg^{-1}=H\}$ is subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ I need to prove that $N(H)$ is subgroup of $G$. It's almost the same question like :$\forall g \in G, gHg^{-1} = H ...
1
vote
0answers
32 views

Let $H$ is a subgroup of $G$. When $G=HZ(G)$? [on hold]

Let $H$ is a subgroup of $G$. found conditions that $G=HZ(G)$?
1
vote
1answer
21 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
2
votes
1answer
73 views

Trirectangular tetrahedron

Looking at http://mathworld.wolfram.com/TrirectangularTetrahedron.html I wonder what the symmetry group of a trirectangular tetrahedron is?
1
vote
1answer
26 views

Is this proof correct for $p_1\times p_2 \mod k = p_3$?

I am trying to prove that if $\gcd(p_1, k) = \gcd(p_2, k) = 1$, then $$\gcd(r, k) = 1$$ where $r = p_1\times p_2 \mod k$. This fact is essential to guarantee that a unit group $U(n)$ of a group ...
2
votes
3answers
95 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
1
vote
1answer
42 views

covering subsets

Let $A=\left\{ {1, 2, \ldots, n}\right\}$. Let $B$ be the set of all size $m$ subsets of $A$. $B=\left\{{B_1,B_2, \ldots , B_{\binom{n}{m}} } \right\}$, $ |B_i|=m$ then we want to find $k$ subsets ...
5
votes
2answers
114 views

Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
-1
votes
1answer
24 views

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$ [duplicate]

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$
3
votes
1answer
22 views

Product of any two non disjoint cycles

Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to ...
0
votes
2answers
45 views

Is it possible for an element of a multiplicative group to have undefined order?

This might be a stupid question but here it is: Let $(G, \phi)$, where $\phi :G \times G \rightarrow G$, be a multiplicative group with identity element $e$. Then is it possible that $\exists a \in ...
1
vote
0answers
19 views

Equivalence of irreducible representations of special linear group SL$(n)$ via those of GL$(n)$ and invariant total anti-symmetric tensor

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. Theorem 13.14 discusses the the equivalence of irreducible representations of special linear group ...
5
votes
0answers
79 views

Aut(G) is abelian

I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ...
3
votes
1answer
37 views

$a,b,N$ are integers. Prove $x=x_0+\cdots$, $\ \ y=y_0+\cdots $ are solutions to $ax+by=N$

I'm asked to prove that if $a,b,N$ are integers, then in the equation: $$ax+by=N$$ I must prove that the integers $$x=x_0+\frac{b}{d}t,\ y=y_0-\frac{a}{d}t$$ are solutions to the equation. where ...
30
votes
3answers
1k views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
4
votes
1answer
119 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
-1
votes
1answer
18 views

Classifying the central product HK of two cyclic groups [on hold]

Let group $H$ be a direct product of cyclic groups $C_1$ and $C_2$ of order $p$ and $p^2$ respectively. Let $D=\{x\in H\mid \text{ord}(x)\leq p \}$. D is generated by $C_1$ and subgroup $E$ of $C_2$ ...
0
votes
0answers
15 views

Finding efficiently finite groups whose set of commutators is not a subgroup

The set of commutators of a group might not be a subgroup of the group. I give here such an example from P.J. Cassidy. It is an infinite group. It is possible to derive from this example one of a ...
0
votes
1answer
27 views

all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
1
vote
2answers
63 views

The center of a group $G$ is a subgroup of $G$ [duplicate]

Definition for the center of a Group: The center $Z(G)$ of a group $G$ is the subset of elements in $G$ that commute with every other element of $G$. Theorem: The center of a group $G$ is a ...
0
votes
0answers
38 views

I want to know if the below sentence is true and why?

I want to know if the below sentence is true and why? Let $G$ be an insoluble finite group then there exists $\pi\subset\pi(G)$ such that if $K=O_{\pi}(G)$ and $\bar{G}=G/K$ it follows that ...
6
votes
5answers
146 views

An example of a group such that $G \cong G \times G$

I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?
0
votes
1answer
26 views

Examine if $\phi$ is a homomorphism and determin $\ker \phi$ and $Im \phi$.

Let $G=(\mathbb C^*,\cdot), G'=(\mathbb R^*,\cdot)$ and $\phi : G\to G'$ be defined by $\phi(z)=|z|, z\in \mathbf C^*$, where $\mathbf C^*=\mathbf C-\{0\}$ and $\mathbf R^+$ is a set of all positive ...
-1
votes
5answers
55 views

If $A$ and $B$ are subgroups of a group $G$, is their product also a subgroup of $G$?

Is there a necessary and sufficient condition for $AB = \{ab|\;a\in A, \; b\in B\}$ to be a subgroup of a group $G$, given $A, B$ are subgroups of $G$?