The study of symmetry: groups, subgroups, homomorphisms, group actions.

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7
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3answers
1k views

Order of the centralizer of a permutation

Given a permutation $\sigma\in S_{n}$, is there a way to know the order of the centraliser $C_{S_{n}}\left(\sigma\right)=\left\{ \pi\in S_{n},\,\pi\sigma=\sigma\pi\right\}$ , i.e what is ...
0
votes
1answer
94 views

What does regular and corefree mean (transitive permutation group)?

I have to prove that a transitive permutation group, $G$, is regular. What is the definition of regular? In addition, my lecturer hinted that a transitive permutation group is regular if and only if ...
7
votes
2answers
387 views

Why is $PGL(2,4)$ isomorphic to $A_5$

In the tradition of this question, why is $\operatorname{PGL}(2,4)\cong A_5$?
5
votes
3answers
536 views

Why prove that multiplicative functions are a group with Dirichlet convolution?

Everyone likes to prove that Dirichlet convolution is a group operation on the multiplicative arithmetic functions, but what consequence does this have? Does any important theorem use this fact? Can ...
6
votes
1answer
426 views

No group of order 400 is simple

I've been given the question of showing that no group of order 400 is simple. I've tried to attack it via the Sylow theorems for about a week now, but all the tricks and methods I know seem to be ...
4
votes
1answer
515 views

Finite abelian groups - direct sum of cyclic subgroup

Let $G$ be a finite abelian $p$-group. It is quite elementary to see that if $g \in G$ is an element of maximal order (and thus its span is a cyclic subgroup of $G$ of maximal order) then $G$ can be ...
1
vote
2answers
334 views

Are these group presentations trivial?

I just got these presentations of groups: $\langle a,b\mid aba^{-1}b^{-1}\rangle$ $\langle a,b\mid aba^{-1}b^{-2},bab^{-1}a^{-2}\rangle$ $\langle a,b\mid abab^{-1}\rangle $ Are any of them ...
3
votes
1answer
82 views

Nonzero element of minimal length in a lattice?

Given two linearly independent vectors $a,b\in\mathbb{R}^2$ we form the lattice $L=\{ma+nb|m,n\in\mathbb{Z}\}$. Now, a proof starts with "choose a nonzero vector in $L$ of smallest length...". Why ...
4
votes
0answers
92 views

Looking for: a subgroup of an uncountable simple group of countable index

Consider the simple group $A_\lambda$, the alternating group on the set $\lambda$, which I will assume has regular cardinality. Recall that this is the smallest subgroup of all permutations of ...
4
votes
0answers
187 views

“Semidirect product” of graphs?

The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd ...
0
votes
1answer
237 views

Algebraic Structures Question

I am having problems understanding what this question is asking. any help would be appreciated. Thanks. The dihedral group D8 is an 8 -element subgroup of the 24 -element symmetric group S4 . Write ...
0
votes
1answer
228 views

Subgroup of discrete group

Let $G$ be a discrete subgroup of $Iso(\mathbb R^2)$. Show that every subgroup of $G$ is discrete. Is it enough to say that since any element of a subgroup of $G$ is also in $G$ it satisfies the ...
3
votes
2answers
135 views

Normal subgroup of automorphisms of a free group

Let $F_2=\langle X,Y\rangle$ be the free group of rank $2$ and consider $A,B,C\in Aut(F_2)$ given by: $$A(X,Y)\mapsto(YX^{-1}Y^{-1},Y^{-1})$$ $$B(X,Y)\mapsto(X^{-1},Y^{-1})$$ ...
6
votes
2answers
223 views

Why do elements of coprime orders commute in nilpotent groups?

I want to show the following statement: Let $G$ be a nilpotent group and $a,b\in G$ such that there exist $m,n\in\mathbf{N}_{>0}$ such that $\text{gcd}(m,n)=1$ and $a^m=b^n=1$. Then $ab=ba$. ...
2
votes
1answer
2k views

Subgroup of index 2 is Normal

Please excuse the selfishness of the following question: Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal. Proof: Because $|G:H|=2$, $G = H \cup aH$ for some $a \in ...
3
votes
1answer
58 views

Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?

I'm having some trouble understanding cosets. If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right? In an exercise, I'm asked the following: ...
0
votes
1answer
249 views

Torsion-free group with a subgroup of finite index [duplicate]

Possible Duplicate: Torsion-free virtually-Z is Z Let $G$ be a torsion-free group with a subgroup $H$ of finite index isomorphic to $\mathbb{Z}$. Is $G$ isomorphic to $\mathbb{Z}$?
4
votes
3answers
126 views

Even subgroup of a freegroup

Show that the even subgroup of ${\bf F}_2$ is generated by $S= \{x^{2}, xy, xy^{-1}\}$. I could use a little help getting started on this problem. I am new to the idea of free groups.
3
votes
2answers
171 views

Elementary formula for permutations?

Suppose I fix $n$ and let $\sigma_k$ represent the $k$th permutation of $S_n$ with respect to some ordering (whatever ordering might serve my purpose). Is there an elementary formula for ...
3
votes
3answers
154 views

Normal and central subgroups of finite $p$-groups

Suppose $G$ is a group where $|G| = p^k$ where $p$ is a prime and $k\gt 0$. Prove that $|Z(G)| \gt 1$; and If $N$ is a normal subgroup of $G$ of order $p$, then $N$ is contained in $Z(G)$.
1
vote
1answer
219 views

Is the torsion subgroup the sum or the direct sum of $p$-primary components?

This is written on Page 93 of Derek J.S. Robinson's A Course in the Theory of Groups: Let $G$ be an arbitrary abelian group, $T$ its torsion-subgroup. For an arbitray prime $p$, denote $G_p$ as ...
43
votes
6answers
2k views

What kind of “symmetry” is the symmetric group about?

There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
17
votes
7answers
5k views

How to show every subgroup of a cyclic group is cyclic?

I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic ...
6
votes
1answer
443 views

Why do the elements of finite order in a nilpotent group form a subgroup?

I would like to prove the following statement: Let $G$ be a nilpotent group. Then the set of elements of $G$ of finite order is a subgroup of $G$. I have no idea but the straightforward ...
6
votes
1answer
115 views

Products of sets in a group

Let $S$ and $T$ be non-empty subsets of a group $G$. As usual $ST=\{st : s \in S, t\in T\}$. What can be said of the subgroups $\langle ST\rangle$ and $\langle TS\rangle$? For example if the identity ...
5
votes
1answer
301 views

Step by step procedure to obtain irreducible representations and construct character table of a group

I am studying group theory and character table of $S_2$ is given in the book. But how to obtain this table is not given. Can someone explain how exactly to construct this table?
1
vote
0answers
85 views

Expressing a permutation in term of its inversions

For $1 \le i \lt n$, let $m_i$ be the number of inversions $(i,j), i \lt j \le n,$ in permutation $ \sigma$. Let $ \sigma_i = (i+m_i,i+m_i-1)..(i+1,i)$. How to prove that ...
3
votes
2answers
136 views

what is exactly the index of a subgroup?

If we have two groups $H$ and $G$ such that $H$ is a subgroup of $G$ we define the index of $H$ in G by the number of all coset of the form $gH$ when $g$ describes $G$ , but I am confused , because ...
5
votes
1answer
204 views

$\operatorname{Hom}(Z_p, Z_p) = Z_p$?

Is $\operatorname{Hom}(\mathbf{Z}_p, \mathbf{Z}_p) = \mathbf{Z}_p$? My proof: $\operatorname{Hom}(\mathbf{Z}_p, \mathbf{Z}_p) = \operatorname{Hom}(\mathbf{Z}_p, \varprojlim\mathbf{Z}/p^n) = ...
5
votes
1answer
73 views

Existence of a subgroup of a certain order

Suppose a group $G$ with $|G|=162$ has a normal subgroup $K$ with $|K|=9$. Show that $G$ must have a subgroup of order 18, using $HK/K \cong H/(K \cap H)$. I feel like this should be very easy, and ...
3
votes
2answers
619 views

Group under matrix multiplication

I am trying to show that this set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$} is a group under matrix ...
0
votes
1answer
94 views

Lie group multiplication/Parameter space

So I have a set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$}. I needed to show P is a Lie group, which I have ...
7
votes
1answer
270 views

Finding 'verbally smallest' element of a finitely generated group

Let $G = ({\Large\ast}^n\mathbb{Z})/K$ be a group, and for each $g \in G$ define $l(g)$ as the smallest positive integer $m$ such that $g = g_1 \ldots g_m$, where each $g_i$ is a generator of $G$. Now ...
2
votes
1answer
299 views

Group action on a subset

having trouble with this problem. It's homework. We're given a finite set $S$ on which a finite group $G$ acts on transitively. If $U$ is a subset of $S$, I'm supposed to show that the subsets of $gU$ ...
2
votes
2answers
460 views

Group, subgroup, homomorphism and core

I've read in three different books about groups, subgroups and homomorphism but I just can't get the concept. I don't find the examples clear enough. For example, why isn't $$\Phi: \mathbb{Z} ...
11
votes
2answers
367 views

Only a solvable group can have 3 Sylow p-subgroups

Is there an easy proof that amongst finite groups, only a solvable group can have exactly 3 Sylow p-subgroups? I have a proof, but it is a bit complex, and I'm looking to use this as an easy ...
3
votes
4answers
204 views

$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: ...
12
votes
3answers
727 views

Pontryagin duality for finite groups

Let $\mathbb{Q}$ denote the group of rational numbers (with addition as the binary operation) and let $\mathbb{Z}$ denote the subgroup of integers. The Pontryagin dual of a group $G$ is the group $G^* ...
6
votes
1answer
187 views

Intersection of neighborhoods of 0. Subgroup?

Repeating for my exam in commutative algebra. Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma: ...
7
votes
1answer
146 views

“indecomposability” in theory of groups and topological spaces

One can define the notion of "indecomposable" in many of the categories that mathematicians think about. However there's no real reason, as far as I can see, to expect it to behave at all well. Here ...
3
votes
1answer
29 views

Why $b a$ and $a^2$ aren't linked in the Cycle graph of Dihedral group $\operatorname{Dih}_4$?

Below is the cycle graph of $\operatorname{Dih}_4$. What I don't understand is that, since $(ba)^2=a^2$, why there isn't a link between $ba$ and $a^2$, and hence of course also $a^2$ and $ba^3$? I can ...
15
votes
2answers
217 views

Help deriving that $\mathrm{sign} : S_n\to \{\pm 1\}$ is multiplicative

$\def\sign{\operatorname{sign}}$ For homework, I am trying to show that $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ we have $$\sign(\sigma_1 ...
0
votes
1answer
231 views

A bijection between automorphisms of a cyclic group and a multiplicative group

This is one more homework question I have--just another one I'm having trouble getting started with. I'm supposed to prove a bijection between the multiplicative group of integers mod $p$ and the ...
8
votes
0answers
666 views

Fundamental group of a compact manifold

In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $$\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$$ where ...
1
vote
3answers
2k views

What's an easy way of proving a subgroup is normal?

I think in most situations(for example, in $S_n$ or $D_n$), proving by definition is too complicated because you have to calculate $gng^{-1}$ for every $n$ in $N$ and $g$ in $G$. To prove that all the ...
3
votes
3answers
265 views

On finite groups

Let $(G,*)$ is a finite group, whose order is $n$. Also let $p|n$ and $p\in P$. Can you bring an example of such group which doesn't have an element of order $p$. Sincerely,
8
votes
1answer
164 views

In a group, does $[G:H] \leq |N|$ always imply $[G:N] \leq|H|$?

Let $G$ be a group and let $H$ and $N$ be subgroups of $G$. Suppose that $[G:H] \leq |N|$. Does this always imply that $[G:N] \leq |H|\ $? Lagrange's theorem tells us that this is true in the finite ...
2
votes
1answer
249 views

Free group contains no notrivial elements of finite order

A free group contains no notrivial elements of finite order This statement seems obvious and trivial, but I cannot think of a nice proof besides going and getting my hands dirty with elements of ...
2
votes
1answer
95 views

What is the “conjugacy problem for differentiable maps”?

A couple of days ago our professor reviewed some of the exercises we had to do and one of them involved giving an example of a conjugacy class in a group. Someone gave an example that involved ...
1
vote
1answer
124 views

Multiplicative Group $(\mathbb{Z}/p\mathbb{Z})^{*}$

Can you give me an example of generator of multiplicative group $$(\mathbb{Z}/p\mathbb{Z})^{*}=\{1, 2, \ldots, p-1\}.$$ Thanks.