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

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747 views

Group theory text [duplicate]

Possible Duplicate: Introductory Group theory textbook I am an Indian student currently in the eleventh grade.I haven't yet learned calculus(I am learning it ) but I would also like to ...
6
votes
7answers
2k views

Group Multiplication Table

I'm currently trying to learn abstract algebra myself, and the following is a quote from the book I am using, "A set of equations, involving only the generators and their inverses, is called a set of ...
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1answer
398 views

Additive Cyclic Group

I'm trying to find an element $k$ that generates the cyclic additive group $\mathbb{Z}_{6}$. Since a group is cyclic, the entire group can be generated by a single element. I've tried adding ...
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votes
3answers
1k views

subgroup generated by two subgroups

Let $A$ and $B$ be two subgroups of the same group $G$. let $$AB=\{ab\,|\, a\in A,\, b\in B \}$$ and $$\langle A,B\rangle$$ the subgroup generated by $A$ and $B$. Are $AB$ and $\langle A,B\rangle$ ...
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1answer
684 views

normalized subgroup by another subgroup

Let $A$ and $B$ be two subgroups of the same group $G$. What does it mean for the subgroup $A$ to be normalized by the subgroup $B$?
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2answers
721 views

$M$ is a maximal normal subgroup iff $G/M$ is simple

$M$ is a maximal normal subgroup of G if and only if $G/M$ is simple. I have a problem in "if" part. To prove ($\Leftarrow$) direction, assume that $N$ is a normal subgroup of $G$ properly ...
3
votes
3answers
262 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
0
votes
1answer
288 views

a definition of semi direct product

let $G$ be a group. and $A$, $B$ be two subgroups of $G$. suppose we have an action of $B$ on $A$ : $\phi:B\rightarrow Aut(A)$ then we can turn the set $AB$ into a group by defining the multiplication ...
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votes
2answers
205 views

uniqueness of a complement of a subgroup

let $A$ and $B$ be two subgroups of $G$. we say that $B$ is a complement of $A$ if : $G=AB$ $A\cap B=\{1\}$ Given a subgroup $A$ of $G$ i don't see how the complement $B$ of $A$ in $G$ is not ...
2
votes
2answers
314 views

Powers and Roots of Group Elements

Let $G$ be a group, and $a$, $b \in G$ $(bab^{-1})^{n} = ba^{n}b^{-1}$, for every positive integer $n$ \begin{align*} \text{Let P(n) be the statement: } (bab^{-1})^{n} &= ba^{n}b^{-1} \newline ...
3
votes
3answers
172 views

Do matrices with central symmetry form a group?

Consider the set of $N\times N$ matrices that satisfy the property $$\mathcal{H} = \{H\,|\, H_{ij}=H_{N+1-i,N+1-j}, \det H \neq 0\}$$ or in matrix forms $$\begin{pmatrix}a_{1} & a_{2} & \cdots ...
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1answer
222 views

Representation which have no unique decomposition into irreducible

What kind of examples of groups and representations should I keep in mind, which do not uniquely decompose into irreducible representations? I am mostly interested in characteristic zero ...
1
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1answer
152 views

Schwartz kernel theorem for induced representation/ Schur algebra for locally compact groups

Given a finite group $G$ and subgroups $H$ and $K$, and representation $$\sigma: H \rightarrow GL(V_\sigma), \qquad \pi: K \rightarrow GL(V_\pi).$$ Now consider the space of functions $f: G ...
3
votes
2answers
137 views

A question about free groups and extensions

Let Nil be the subgroup of $GL_3(\mathbb{R})$ given by matrices of the form $$ \left( \begin{array} 11 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1 \end{array} \right) $$ with ...
2
votes
4answers
467 views

Connection Between Automorphism Groups of a Graph and its Line Graph

First, the specific case I'm trying to handle is this: I have the graph $\Gamma = K_{4,4}$. I understand that its automorphism group is the wreath product of $S_4 \wr S_2$ and thus it is a group of ...
3
votes
2answers
365 views

Coloring dodecahedron

I found some months ago that there are the Polya's enumeration theorem to compute number of colorings of dodecahedron. I got interested to find how to show by using only Burnside's lemma that there ...
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1answer
231 views

Functor from a group to a poset

Are there examples of functors from the category of a single group to the category a partially ordered set (some sort of representation of the group in a poset) ?
3
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0answers
188 views

Automorphism Group of Paley Graph

I would like an explanation as to the structure description of the automorphism group of a Paley graph. Paley graphs are a specific case of Cayley graphs where the group is $Z_q$ (q is a prime power ...
5
votes
1answer
277 views

Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?

Can anyone give me a reference to the aforementioned theorem? W. Hodges uses it for an example in his "Model Theory", but I couldn't find anything on it yet. The group may be (let's say, countably) ...
4
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1answer
194 views

Subgroups of the braid groups $\mathcal{B}_n$

Are the braid groups $\mathcal{B}_n$ virtually abelian ? virtually free ?
3
votes
3answers
189 views

The product of all the elements of a finite abelian group

I'm trying to prove the following statements. Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, ..., a_{n}\}$. If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} ...
0
votes
1answer
100 views

Subgroups with conjugate closures in a profinite group

Suppose $G$ is a profinite group and $H$ and $K$ are subgroups with conjugate closures. Does it follow that $H$ and $K$ themselves are conjugate in $G$?
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votes
2answers
139 views

Group acting nilpotently on another group

Suppose $G$ is a group acting faithfully by automorphisms on a group $K$, and let $[k,g]=k^{-1}k^g$ for $k\in K$ and $g\in G$. The subgroup generated by these elements we'll call $[K,G]_1$, and we ...
2
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1answer
446 views

Proving that (group) automorphisms form a subgroup of the permutation group on the set

I've to prove that $(\text{Aut}(G), \circ)$ is a subgroup of $(\text{Perm}(G),\circ)$ where: $G$ is a group over the set $S$. $\text{Perm}(G)$ denotes the set of all permutations in $S$ ...
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5answers
607 views

Finite Abelian Group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2}$ = $e$. I've been stuck on this problem for quite some time. Could ...
2
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1answer
526 views

Counting Elements and Their Inverses

The problem I am attempting to prove is the following: In any finite group $G$, the number of elements not equal to their own inverse is an even number. Caveat: I have had very limited experience ...
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votes
4answers
4k views

Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$

The following question is problem Pinter's Abstract Algebra. And to put things in context: $G$ is a group and $a, b$ are elements of $G$. I want to show $(ab)^{-1}$ = $b^{-1}a^{-1}$. I originally ...
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3answers
365 views

Matrices of irreducible representations of common groups

I was wondering where can one find the matrices (and not just the character tables) of the irreducible representations of the most common groups (alternating, symmetric, octahedral, etc..) ? Thanks ...
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votes
3answers
560 views

Distinguishable painted prisms with six colors (repetition allowed)

Fraleigh(7th) Ex17.9: A rectangular prism 2 ft long with 1-ft square ends is to have each of its six faces painted with one of six possible colors. How many distinguishable painted prisms are ...
3
votes
2answers
268 views

Showing $\mathbb C G$ modules are projective

Let $\mathbb C G$ be the group ring of a finite group $G$ and let $V$ be an irreducible $\mathbb C G$ module. I'm having trouble showing that $V$ is projective. Any help? Thanks!
2
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1answer
90 views

What's an example of a group with equivalent uniform structures where multiplication is not uniformly continuous?

Say we have a topological group $G$. It's easy to see that if $\cdot: G \times G \rightarrow G$ is uniformly continuous (with respect to either the right or left uniformity), then $G$ must have ...
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2answers
1k views

Primitive root and discrete logarithm

Started studying Diffie Hellman key exchange protocol as part of the course. I dont have much maths knowledge. Can somebody explain in simple terms what is the significance of taking Primitive root in ...
3
votes
1answer
121 views

Fourier analysis on groups and paths in a Cayley graph

If one takes a cyclic group and a function on this group, and performs harmonic analysis on it (classical Fourier analysis), the result is a set of coefficients, each one of them corresponding to ...
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votes
4answers
3k views

Finite Groups with exactly $n$ conjugacy classes $(n=2,3,…)$

I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes. If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of ...
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1answer
401 views

Why the discrete subgroup of $R^n$ has finite balls?

Consider a discrete subgroup $G$ of $R^n$. Then let we have an open ball $B\subset G$ (topology in $G$ is induced from $R^n$). Why is $|B|<\infty$? For example I can take ...
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1answer
143 views

Ways of building groups besides direct, semidirect products?

Let's say we have a group G containing a normal subgroup H. What are the possible relationships we can have between G, H, and G/H? Looking at groups of small order, it seems to always be the case that ...
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6answers
762 views

Why are projective objects important?

I belive we study them because in important categories they are close to free objects and even a retract of a free object in some algebraic instances (for example, direct summands in Mod_R, and ...
2
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1answer
337 views

If $F(a, b)=\langle a, B\rangle$ then $B=a^ib^{\epsilon}a^j$: a neat proof?

If you take a generating pair for $F(a, b)$, $(a, B)$, then it is intuitively obvious that $B=a^ib^{\epsilon}a^j$ for $i, j\in \mathbb{Z}$, $\epsilon=\pm 1$. However, I cannot come up with a neat ...
2
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1answer
79 views

Length of a not normal subgroup

For a group $G$ the length $l(G)$ is defined as the supremum of the lengths of all composition series of $G$. If $N$ is a normal subgroup of $G$, then $l(G)=l(N)+l(G/N)$. This, I believe, is a ...
3
votes
1answer
195 views

Generating elements of the free group of rank two up to conjugacy

I've tried to google how to generate words in two generators up to conjugacy (this means I only want one representative for each conjugacy class). Sadly, I come up with articles that have, in the ...
8
votes
4answers
246 views

What property of groups is related to the symmetries?

In many elementary group theory books there are pictures of symetries for objects ( Cube, Square etc. ) and some group that represents that symmetries. My question is what property of a group is ...
3
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0answers
224 views

[H,K] abelian if K centralizes [H,K]

Here is a paraphrased version of problem 4B.4 in Isaacs's Finite Group Theory: Let $G$ be a group and $X,Y$ subgroups of $G$, such that $Y$ centralizes $[X,Y]$. If $X$ is normal in $G$, show ...
3
votes
2answers
483 views

Every finite group of isometries of the plane is isomorphic to $Z_n$ or $D_n$

Fraleigh(7th) Theorem 12.5: Every finite group $G$ of isometries of the plane is isomorphic to either $Z_n$ or to a dihedral group $D_n$ for some positive integer $n$. (Note: An isometry of ...
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1answer
203 views

Finding all results of a permutation group on a set

Given a finite group $G < Sym(\Omega)$; $\Omega$ finite, and $X \subset \Omega$, I can define a by the function $H(g) = \{x^g \| x \in X\}$ for each $g \in G$. Of course, each $H$ has the same ...
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0answers
346 views

Irreducible finite dimensional representations of $SL(2,\mathbb C)$

Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$? For example, how can I show "directly" ...
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0answers
64 views

Lie algebra of a group and its profinite completion (reference request)

Let $G$ be a group and let $G_n$ be its series of dimension subgroups defined as follows: $$G_n=\{g \in G | g-1 \in \Delta\} $$ where $\Delta$ is the augmentation ideal. This series has the property ...
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2answers
461 views

Induction from normal subgroups

Let $G$ be a (finite) group and $N$ a normal subgroup. Given an irreducible representation $\pi$, how can I decompose $Ind_N^G \pi$? I'd be happy also about a good reference for this.
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1answer
564 views

every transitive action of an abelian group is regular

Why the following is true : "Every transitive action of an abelian group is regular" Does this mean that every action of an abelian group is free? because as i understand, a regular action is ...
2
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2answers
179 views

Understanding some strange notation for $D_4$

I'm now studying Fraleigh's Abstract algebra(7th). In section 8, there is a group table for $D_4$, with some strange notations that I can't compute it easily. He uses $\rho_0=\left( ...
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6answers
2k views

If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic

Fraleigh(7th ed) Sec10, Ex47. Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^m=e$ in $G$ is at most $m$, then $G$ is cyclic. ...