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
2answers
34 views

What does it means to multiply a permutation by a cycle? $\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$

I have to prove that $$\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$$ but I can't understand what this means. My book doesn't defines what a permutation and cycle product would be. So, for ...
1
vote
1answer
16 views

Finding Automorphisms of Irregular graph through Regular Sub-Graphs.

Objective : To find a set of permutations for a irregular graph which is also a set of automorphism. This finding process uses permutations of 2 regular subgraphs of the given graph. Description and ...
2
votes
1answer
38 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 ...
1
vote
0answers
40 views

A question about semidirect product

When we consider the classification of the group G by semidirect product, we need to consider all the homomorphisms from K to Aut(H), Where G=HK and H$\unlhd$G,H$\bigcap$K=1 But by the theorem: ...
3
votes
1answer
42 views

Insight about compact groups

I'm quite familiar with the general notion of compactness in math but I have some troubles with its extension to group theory. I'm not talking about definitions or theorems: I would like to have some ...
1
vote
1answer
16 views

If $s$ and $t$ are symmetries of a plane such that they agree on three non collinear points then show that $s=t$

This is a problem based on "Symmetry" of the plane $\mathbb{R^2}$. Suppose $A$, $B$, $C$ are the three points in plane which are after the corresponding actions by $s$ and $t$ are in the places $D$, ...
1
vote
1answer
41 views

Number of Automorphisms of a Irregular Graph.

I have been looking for results on number of graph automorphisms of irregular graph(upper and lower bound). I searched , but could not find anything which can be used directly. Say, $G$ is $k$ ...
12
votes
2answers
111 views

Why this $\sigma \pi \sigma^{-1}$ keeps apearing in my group theory book? (cycle decomposition)

I'm studying cycle decomposition in group theory. The exercises on my book keep saying things like: Find a permutation such that $\sigma (1 2) \sigma^{-1} = (123)$ Prove that there is no permutation ...
0
votes
1answer
32 views

Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
2
votes
0answers
38 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic groups ...
2
votes
1answer
31 views

Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
-7
votes
0answers
35 views

Show that $Nil(\mathbb{Z}_n) $ is a subgroup of $\mathbb{Z}_n$ [on hold]

Show that $\mathrm{Nil}(\mathbb{Z}_n) = \{\bar{x}\in \mathbb{Z}_n\mid \bar{x}\,^m=\bar{0}\text{ for some positive integer $m$}\}$ is a subgroup of $\mathbb{Z}_n$.
1
vote
1answer
26 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
0
votes
1answer
69 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
13
votes
0answers
74 views

Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...
-2
votes
0answers
23 views

Proofs of Sylow Theorems [on hold]

‎let ‎‎$‎G‎$‎‏ ‎is a‎‎ ‎finite‎ group and ‎$‎‏p‎$‎ is prime. if ‎$‎P‎\in Sly‎_{p }(G)‎$‎‏‎‎ then ‎$‎O‎_{p}(G)=Core‎_{G}(P)‎$‎‏‎‎
-1
votes
0answers
33 views

problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
3
votes
2answers
50 views

Permutations minus Transpositions

I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition. Attempt: Lets call $g(n)$ the ...
2
votes
1answer
57 views

What finitely generated amenable groups are known to be LERF?

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated"). I'm looking for examples (many, if possible) of groups which are: Finitely generated, but infinite ...
4
votes
3answers
157 views

Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
0
votes
1answer
23 views

How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
0
votes
1answer
31 views

Bijection from conjugacy class to the factor group by centralizer.

How different is $g^{-1}xg$ from $gxg^{-1}?$ Because proving a bijection from $g^{-1}xg$ type conjugacy class to the set of right cosets of the centralizer of $g$ in $G$ is as easy as proving it from ...
1
vote
2answers
44 views

Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
0
votes
0answers
67 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
4
votes
2answers
48 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
0
votes
0answers
22 views

Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix. The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots ...
0
votes
0answers
25 views

Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
3
votes
2answers
34 views

Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
3
votes
0answers
73 views

$Ker (f)$ is finite, then $G$ is finite. [on hold]

Let $G$ be a group with identity element $e$, $f: G → G$ a homomorphism for which there is a natural $n> 1$ such that $f^n (G)$ = {e}. i. Prove that if $Ker ...
0
votes
3answers
53 views

Does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$?

I know that $G\cong G'$ and $H\cong H'$ implies $G\times H\cong G'\times H'$. But is it true for reverse? I mean, does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$? If so, how to ...
-1
votes
1answer
18 views

If a set $X$ contains three different elements $a,b,c$ describe $f:=t(a,b)∘t(b,c)$ and $g:=t(b,c)∘t(a,b)$. Are they equal?

The group of permutations of a set $X$ consists of all functions $f:X\to X$ that are one-to-one and onto. The group operation is the composition of functions. Of special importance are transpositions ...
2
votes
0answers
55 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
0
votes
0answers
29 views

Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?
4
votes
0answers
30 views

Number maximum of commutators required to generate an element of the derived subgroup

Let $G$ be a group for which the center $Z(G)$ is of index $n$. How to prove that an element of the derived subgroup $G^\prime$ is the product of at most $n^3$ commutators?
1
vote
1answer
61 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
3
votes
2answers
84 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
2
votes
1answer
47 views

On group with special properties

Is there a group $G$ with two the following properties:? i) $Aut(G)$ is not nilpotent, where $Aut(G)$ is the full automorphism group of $G$. ii) There exists an element $1\neq x\in G$ of order ...
5
votes
0answers
35 views

A normal subgroup of $ GL(n, K) $

Let $ F $ be a field and $ K $ be an extension of $ F $. Define the set, $$ E(n, K, F) := \{ M \in GL(n, K) , \det M \in F \} $$ Show that $ E(n, K, F) $ is a normal subgroup of $ G(n, K) $ and also ...
-1
votes
0answers
31 views

Orbits of the symmetry group and the alternating group [on hold]

I have difficulties with these problems. Any solutions will be appreciated. 1) Compute the orbits of the symmetric group of the tetrahedron on the set of 6 pairs of vertices. 2) Compute the orbits ...
5
votes
1answer
46 views

Recognizing action of semidirect product

I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The ...
-2
votes
0answers
34 views

Orbits of the tetrahedron [on hold]

Compute the orbits of the symmetry group of the tetrahedron on the set of $6$ pairs of vertices. What if the tetrahedron was an icosahedron with 66 pairs?
-2
votes
0answers
20 views

Hypercyclic Group [on hold]

A group $G$ is Hypercyclic group if any Sylow subgroup of $G$ is cyclic. Can you please give some idea how to solve this Questions? $1)$ find example of hypercyclique group that is not cyclic. $2)$ ...
1
vote
2answers
72 views

Rubik's Cube's Group

Is there an article somewhere with an exhaustive study of the Rubik's Cube Group $G$? Such as computing some subgroups of it or exhibiting some elements of its center $Z(G)$? I tried googling it and ...
1
vote
2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
1
vote
2answers
59 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
1
vote
0answers
42 views

$|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ commute.

Can anyone help me with this exercise? "Let $G$ be a group. Let $N$ be a normal subgroup and $H$ a subgroup of $G$. Assume that $|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ ...
3
votes
0answers
67 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
4
votes
1answer
59 views

Homomorphism from $\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}\rightarrow \mathbb{Z}\oplus \mathbb{Z}$ has non-trivial kernel: elementary argument

One can give an elementary arguments (avoiding "rank") to prove that any group homomorphism $f$ from $\mathbb{Z}\oplus \mathbb{Z}$ to $\mathbb{Z}$ has non-trivial kernel: Let $f:(1,0)\mapsto a$ and ...
0
votes
0answers
44 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...
2
votes
2answers
28 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...