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

learn more… | top users | synonyms (2)

2
votes
1answer
83 views

Verifing some properties about $G$

I have the following problem$^*$: Prove that the group $G$ having generators and relations respectively $$X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$$ is an infinite ...
1
vote
1answer
52 views

$G$ is torsion-free group then $G/\langle X\rangle$ is torsion

Honestly, I have been thinking on this problem for hours but couldn't find a way: Let $G$ is torsion-free group and $X$ is a maximal independent subset, then $G/\langle X\rangle$ is torsion. I ...
0
votes
1answer
51 views

Characterizing minimal non-FC-groups…

Let $G$ be a minimal non-FC-group and suppose $G^*<G$ (where $G^*$ is the finite residual). Then we have (i) $G=<G^*, x>$, $x^{p^n} \in G^*$ and $x^p\in Z(G)$, (ii) $G^*$ is a divisible ...
5
votes
2answers
632 views

Homomorphism between $A_5$ and $A_6$

The problem is to find an injective homomorphism between the alternating groups $A_5$ and $A_6$ such that the image of the homomorphism contains only elements that leave no element of ...
1
vote
2answers
1k views

Are all finite groups cyclic?

I've been reading about Number Theory and I came across this proof that the finite subgroups of the multiplicative group of a field is cyclic. However, it seems the proof applies to all finite groups ...
1
vote
1answer
103 views

Divisible abelian $q$-group of finite rank

What does "finite rank" mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
1
vote
2answers
547 views

Lie Algebra of U(N) and SO(N)

U(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?
2
votes
1answer
235 views

Why is a normal subgroup containing $\Phi (G)$ with a nilpotent factor group nilpotent?

At page (28) of chapter I of the book Finite Group Theory by I.Martin.Issacs, one finds: Let $G$ be a finite group, with Frattini subgroup $\Phi$. If $\Phi \subseteq N \vartriangleleft G$, and ...
0
votes
1answer
60 views

Is there an associative (monoid) operation on $\mathbb{R}_{\geq 0}$ which is also a metric?

Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there ...
6
votes
2answers
251 views

Why is the name general “linear” group?

Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?
7
votes
2answers
214 views

Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?

What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
3
votes
1answer
104 views

Order of $\mathrm{Aut}(G)$

I am not sure how to approach the following problem: Show that if $|G|=n$, then $|\mathrm{Aut}(G)|$ divides $(n-1)!$ All I can think of so far is that clearly $|\mathrm{Aut}(G)| \le ...
1
vote
1answer
67 views

Exact sequence of abelian groups, property transfers

We had the statement that with an exact sequence of multiplicatively written abelian groups $U \mapsto V \mapsto W \mapsto X \mapsto Y$ and in $U$, $V$, $X$, $Y$ every group element has a unique ...
1
vote
4answers
192 views

What is an odd permutation in the centralizer of (1,2,3)(4,5,6) in $S_6$?

Can you help me finding an odd permutation which commutates with $(1,2,3)(4,5,6)$ in $S_6$ ?
8
votes
1answer
2k views

Centralizer of a given element in $S_n$?

It is known that any two disjoint cycles in $S_n$ commutes. Therefore, any $\pi\in S_n$ which is disjoint with $\sigma$ is in the centralizer of $\sigma$: $C_{S_n}(\sigma)$. Also $$ \sigma^i\pi\in ...
1
vote
3answers
2k views

Simple method the show that a group of order 15 is cyclic [duplicate]

How can i show with quite simple methods that a group of order 15 is cyclic?
2
votes
2answers
771 views

Cayley tables for two non-isomorphic groups of order 4.

Not sure how to make tables, but: For a binary operation $*$, and set $\{e,a,b,c\}$, in the Cayley table, $a*a$ can be filled with either the identity or an element different from both $e$ and $a$. ...
2
votes
2answers
62 views

Derived subgroup [U,V] in centre of subgroup generated by U and V

I'm quite new to this whole topic and so I don't know how get a grip on this question: Let $G$ be a group and $U,V$ two subgroups. Denote by $[U,V]$ the subgroup of $G$ generated by ...
1
vote
1answer
262 views

$G$ contains a abelian normal nontrivial subgroup $H$ with $G/H$ has derived length is $n-1$

Let $G$ be finite group. Show that if $G$ is a solvable group, and derived length is $n$ then $G$ contains a abelian normal nontrivial subgroup $H$ with $G/H$ has derived length is $n-1$
0
votes
2answers
110 views

If $G$ is a finite nonsolvable group then $G$ contains a nontrivial subgroup $H$ such that $\left[H,H\right]=H$.

Show that if $G$ is a finite nonsolvable group then $G$ contains a nontrivial subgroup $H$ such that $\left[H,H\right]=H$.
0
votes
2answers
154 views

What does the “direct product” mean in this context?

Every group of order $231$ is the direct product of a group of order $11$ and a group of order $21$. By Sylow's theorem, we know there are one Sylow-7 subgroup, one Sylow-11 subgroup (these two ...
2
votes
2answers
129 views

$T(G)$ may not be a subgroup?

It is obvious that for an abelian group $G$; the set of all torsion elements: $$T(G)=\{x\in G|x^n=0 \text{, for some nonzero integer } n\}$$ is a subgroup of the group. I am asked to probe this fact ...
1
vote
1answer
182 views

Terminology question; inverse vs complement in Boolean algebra

This was said at a lecture I attended: $e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$. So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
1
vote
2answers
332 views

$A_4 \oplus Z_3$ has no subgroup of order 18

Here my solution: Suppose there exists and $H \leq A_4 \oplus Z_3$ such that order of H is 18. Now, notice index of H in $A_4 \oplus Z_3$ is 2. therefore, H is normal, and therefore, the $A_4 \oplus ...
2
votes
1answer
234 views

Conjugacy classes of elements of a prime order in $PSL_2(q)$

Let $q=p^f$ be a prime power. Given a prime number $r$, how many conjugacy classes of elements of order $r$ are there in $PSL_2(q)$? This topic should have appeared in literature, and I am told ...
1
vote
1answer
96 views

Minimal non-FC Groups

Let $G$ be a minimal non-FC-group with $G'<G$. Suppose that $G$ has no non-trivial finite factor group (absurdum hypothesis). Now, $G\over G'$ is a divisible abelian group; but also periodic? ...
2
votes
2answers
140 views

Prove $H \times G$ is commutative iff $H, G$ are commutative

This is another proof question I am asking about, can someone give me tips on how to answer these questions? My question says: "Let $H,G$ be arbitrary groups. Prove that $H \times G$ is commutative ...
0
votes
2answers
73 views

Smallest $m \mid 60$ so that there is no subgroup $H \leq A_5$ with order $m$

What is the smallest $m \mid 60$ so that there is no subgroup $H \leq A_5$ with order $m$ ?
1
vote
3answers
530 views

Calculation of subgroups of $(Z_{12}, +)$

How calculate all subgroups of $(Z_{12}, +)$? I know that the order of subgroups divide the order of the group, but there is such a smart way to calculate the subgroups of order 6?
0
votes
1answer
124 views

Center of a Normal Group

Do the definition of the center of a group apply to subgroups. For example if $N$ is a normal subgroup of $G$, I want to consider $Z(N) = \{ n \in N |\space nx = xn \space\space x \in N\} $. I realize ...
3
votes
1answer
212 views

Maximal Free Subgroups and Torsion

Let $G$ be a non-trivial torsion-free Abelian group. If $T$ is a maximal free subgroup of $G$, then $G\over T$ is periodic? How can we prove this?
1
vote
2answers
126 views

A question on prime power order group.

After some trying on this problem I could not solve it: For a group of order $p^n$, $p$ prime, prove that for any subgroup $H\ne G$, $\exists x\in G, x \notin H$ such that $xHx^{-1}=H$. Can someone ...
6
votes
3answers
285 views

An Example where $gHg^{-1} \ne g^{-1}Hg$

I was trying to think of an example where $gHg^{-1} \ne g^{-1}Hg$. I couldn't think of one, but I am curious if the following reasoning demonstrates that, at the very least, such an example must ...
2
votes
1answer
136 views

Is a subgroup of GL_2(C) a group of order 12?

Consider the subgroup $G$ of $GL_{2}(\mathbb{C})$ generated by $A=\begin{pmatrix} \omega & 0 \\ 0 & \omega^{2} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$ where ...
1
vote
2answers
205 views

How to show a certain map is surjective?

Let $G$ be a group and $H,K \subset G$ be normal subgroups such that $H \cap K = \{e\}$ and $G=HK$. I need to show that the map, which I have denoted $\phi$, $H \times K \longrightarrow G$ given by ...
1
vote
1answer
178 views

Subgroup of rotations in $D_n$ is normal.

Here is what I did: Suppose R $\leq$ $D_n$ is the subgroup of rotation. Suppose $x \in fRf^{-1}$. Then $x = frf^{-1}$ for some r and for any r. Therefore, $x = r'$ a rotation and hence $x \in R$. ...
1
vote
0answers
76 views

Help to show that if this Group is Metabelian?

Here is my problem: Let $G=\langle a,b|a^l=b^3=1,(ab)^3=(a^{-1}b)^3 \rangle$. Find the order of $\frac{G}{G'}$ and then verify that if $G$ is metabelian. What I have done: I added the relation ...
3
votes
2answers
163 views

A question about composition of the inverse image of a group homomorphism and the homomorphism itself

Suppose $\phi:G_1 \rightarrow G_2$ is a group homomorphism and $H \leq G_1 $. Show that $\phi^{-1}(\phi(H))=H \cdot \ker(\phi) $. Attempt at a solution: I was easily able to show that ...
0
votes
2answers
62 views

Subgroup of $D_n$ with order $m$ as $m \mid 2n$

I have to prove that as $m \mid 2n$ then there is a subgroup of $D_n$ with order $m$. How can i do that ?
8
votes
1answer
337 views

An Intuitive Explanation of the Transfer Homomorphism

I just learned about the transfer homomorphism, and I am having trouble internalizing it. I am learning from 'A Course in the Theory of Groups', and I was hoping that perhaps someone had a more ...
1
vote
1answer
285 views

Finding smallest possible group of matrices containing a given matrix

I am trying to understand the process for solving group theory questions. Let $a=\begin{bmatrix} 1&1\\0&1 \end{bmatrix}$ and $b=\begin{bmatrix}i&0\\0&-i\end{bmatrix}$ - 2 x 2 matrices ...
2
votes
3answers
215 views

Help explaining the sign of a permutation

I have a permutation that has been expressed in disjoint cycles (this isn't my actual question, this is an example done in lectures which I'm trying to understand): (a b c)(d e f g h i) Now the ...
1
vote
3answers
460 views

Find a group with four elements in which every element is its own inverse

What is the procedure for solving this problem. It seems like I can just say let $G$ be a group $\{1, a, b ,c\}$ and define $a * a^{-1} = b * b^{-1} = c * c^{-1} = 1$ ... but then what does $a*b$ or ...
0
votes
3answers
529 views

Showing the group of integers modulo m with multiplication is a group

I've just started group theory and I don't know how to show this. Are you supposed to take examples of two elements from the group and an example modulo m (say 2 and 5, and modulo 3) and show the ...
3
votes
0answers
188 views

short exact sequence - split, as a semidirect product, with some cohomology

I've looked at several s.e.s. examples and I feel I am quite close but here is a question I am still a little confused on. Let $E$ be a group and $A$ an abelian normal subgroup s.t. have an exact ...
4
votes
1answer
95 views

Can a nontrivial group contain its operation?

I am searching for examples of groups which contain their own operation as an element. I am having difficulty showing that this is not possible for groups of size greater than 1, but counterexamples ...
1
vote
0answers
137 views

Estimating Number Of normal subgroups of a p-Group

Does someone have an idea about a possible way to count the number of normal subgroup that a group of order $p ^n $ has ($n \in \mathbb{N}$ )? Is there anyway we can count the maximal subgroups it has ...
0
votes
1answer
180 views

Action on a doubly transitive group - induced action

Assume that the group $G$ acts on $X$ . Then define $\forall x,y \in G \forall g \in G: g(x,y) := (g(x),g(y))$, so that $G$ acts on $X^2$ by this action. I have to prove that $G$ has at least two ...
2
votes
1answer
179 views

Maximal subgroups of of the alternating group of degree 5

$A_5$ has the maximal subgroup $S=<(123),(12)(45)>=\{e,(123),(132),(12)(45),(13)(45),(23)(45)\}$. $A_5$ has $6$ maximal subgroups of order $10$ and has $5$ maximal subgroups of order $12$. I ...
2
votes
0answers
53 views

groups acting on curves

Can anybody help me to prove the following statement ? *Let $X$ be a smooth, connected projective curve defined over a number field $k \subset \mathbb{C}$. Let $G$ be a finite group acting on $X$. ...