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

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1answer
51 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 ...
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0answers
153 views

How to solve a system of equations over permutations?

Given a system of equations $P_{1} = x_{11} \cdot x_{12} \cdot \ldots \cdot x_{1n}$ $\ldots$ $P_{l} = x_{l1} \cdot x_{l2} \cdot \ldots \cdot x_{ln}$ where $P_{i} \in S_{n}$ are permutations over ...
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2answers
460 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?
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1answer
92 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? ...
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1answer
57 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 ...
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2answers
209 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
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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 ...
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4answers
180 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 $S_6$ ?
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1answer
64 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 ...
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2answers
61 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 ...
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2answers
105 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 $\left[H,H\right]=H$.
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1answer
228 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 $n-1$
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2answers
267 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 ...
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1answer
130 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 ...
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2answers
72 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 $m$ ?
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1answer
122 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
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1answer
197 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?
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2answers
124 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 ...
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3answers
273 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
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1answer
366 views

Finitely Generated Group

Let be $G$ finitely generated; My question is: Does always exist $H\leq G,H\not=G$ with finite index? Of course if G is finite it is true. But $G$ is infinite?
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3answers
877 views

Seifert-van-Kampen and free product with amalgamation

I would like to apply Seifert van Kampen to a simple example taken from Wikipedia: I have $X = S^2$ and $A = S^2 - n$, where $n$ is the north pole and $B = S^2 - s$, where $s$ is the south pole. ...
3
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1answer
176 views

A question about a group presentation

I have calculated the fundamental group of the annulus and got the following group presentation: $$ \langle a, b | ab = ba = 1 \rangle$$ This is the set of strings of the form: $1, a, a^2, a^3, ...
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1answer
122 views

Question about the sum of chain groups

What's the difference between $C_n(A + B)$ (that is, $C_n(A)+C_n(B)$ in $C_n(X)$) and $C_n(A) \oplus C_n(B)$ where $A$ and $B$ are subspaces of a topological space $X$? They're the same sets, right? ...
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2answers
192 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 ...
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1answer
130 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 $x \in R$. ...
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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 ...
7
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0answers
186 views

A problem in transfer theory

This is about problem 5B.1 page 157 in Isaacs' "Finite Group Theory" book. This chapter is definitely giving me trouble. The problem reads: Let $G$ be a finite group, $P \in \operatorname{Syl}_p(G)$ ...
7
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1answer
255 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 ...
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1answer
428 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
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2answers
152 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 ...
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2answers
522 views

A weird proposition in Lang's “Algebra”

It goes like this: Let $G$ be a group, $H < G$, and $N \lhd G$ be the smallest normal subgroup containing $H$. Then any $f \in \operatorname{Hom}(G,G')$ such that $H$ is in its kernel uniquely ...
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2answers
191 views

existence of homomorphisms between free products of groups

If there exists a homomorphism $f: A \rightarrow B$ between two groups $A$ and $B$, and a homomorphism $g: C \rightarrow D$ between two groups $C$ and $D$, then will there exist a homomorphism $h: A*C ...
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2answers
245 views

Is the natural action of $S_n$ on $\{1, 2, \dots, n\}$ a left or a right action?

The symmetric group $S_n$ acts naturally on the set $\{1,2,\dots, n\}$. Is this a left or a right action?
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5answers
325 views

A criterion for a group to be abelian

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let and ...
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1answer
268 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 ...
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3answers
191 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 ...
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0answers
175 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 ...
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3answers
399 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 ...
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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 ...
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0answers
126 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
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1answer
143 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 ...
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1answer
171 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 ...
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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 $X$. ...
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1answer
96 views

How to show that $G\cong\mathbb Z_3$

I am working on this one: Let $G=\langle x,y|xyx^{-1}y^2=1, yx^{-1}yx^2y=1\rangle$. Show that $G\cong\mathbb Z_3$ What I have done for this is to consider subgroup $H=\langle x\rangle$ and to ...
3
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2answers
185 views

Group Theory - Normal Subgroups

Let $G$ be a finite group and $N$ a normal subgroup such that $|N|$ and $|G/N|$ are relatively prime. (i) Let $H$ be a subgroup with the same order as $G/N$; prove that $G=HN$. (ii) Let $g$ be an ...
2
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2answers
155 views

Induced automorphism on quotient group

I am stuck on a problem studying for my prelims : Let $G$ be a group, $N \unlhd G$ a normal subgroup and $\alpha \colon G \to G$ an automorphism of $G$ such that $\alpha(N) \leq N$. The first part is ...
2
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3answers
226 views

Surjectivity of product map

Let $H$ be a subgroup of a group $G$, let $\phi: G \rightarrow H$ be a homomorphism with kernel $N$, and suppose that the restriction of $\phi$ to $H$ is the identity. I am trying to determine ...
3
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2answers
77 views

What is the inverse of this map?

Let $G$ be a group. Suppose $f:G\to G/N$ is the canonical map, where $N=\ker f $ is normal in $G$. I wonder what $f^{-1}(G/N)$ is explicitly in this context. I see some sources say it is a disjoint ...
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2answers
76 views

How to prove the equivalence of definitions of doubly transitive group

Given a group $G$ which acts on a set $X$ we define that a group $G$ acts doubly transitive on $X$ as $G$ is transitive on $X$ and $G_x$ is transtive on $X \backslash \{x\}$ where $G_x$ the stabilizer ...
0
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1answer
134 views

Explicit calculation to show a complex is exact

I know that the image of the preceding map, when equaling the kernel of the next map, means the complex is exact. Furthermore, this operator, listed below, is linear, but I am having a hard time ...