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

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2
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63 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
4
votes
1answer
35 views

Proving the Thompson Transfer Lemma

Let $G$ be a finite group of even order $n=2^kr$, $T$ a Sylow-$2$ subgroup of $G$, and $M$ an index $2$ subgroup of $T$. I want to show that if $G$ has no subgroup of index $2$, then every element $x$ ...
0
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0answers
32 views

An error in least square optimization problem in Matlab

I am new to MATLAB and I want to formulate the following lease square expression in Matlab. I have some codes that I am typing here. But the optimization problem solution seems not to be correct. Does ...
0
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0answers
32 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
2
votes
1answer
48 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
2
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2answers
44 views

I.N. Herstein, “Topics in algebra” group theory section 2.8 example 2.8.1

I.N. Herstein, "Topics in algebra" group theory section 2.8 example 2.8.1 it is written that Let $G$ be a finite cyclic group of order $r$, $G=(a)$, $a^r=e$. Suppose $T$ is an automorphism of $G$. If ...
6
votes
1answer
48 views

Definition of the normalizer of a subgroup

Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion $N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$? Thanks!
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2answers
112 views

Help with semidirect product

I need help with this problem, i am trying to understand the semidirect product, so if anyine could help or give me some ideas Let $G$ be the group generated by $<a,b>$ and the relations ...
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3answers
51 views

Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
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0answers
24 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
0
votes
1answer
69 views

a potential application of the ping-pong lemma?

From my understanding, a simple result of the ping-pong lemma would state that if we have a set of linear transformations (matrices) $A_1,\ldots,A_n$ all of the same dimension, then if ...
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0answers
29 views

Can a compact topological group have the trivial topology?

I want to show that the following are equivalent for a compact topological group $G$: $G$ is the inverse limit of finite groups $G_i$. There's a family $\left\{N_i\right\}$ of open normal subgroups ...
3
votes
0answers
68 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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vote
0answers
51 views

Order Preserving Isomorphism

If abelian group G has an archimedean order then there is an order preserving isomorphism $\phi$ of G onto a subgroup of $\mathbb{R}$. Here we can say that G is archimedean totally ordered abelian ...
2
votes
1answer
55 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
9
votes
2answers
102 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
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0answers
30 views

representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
7
votes
1answer
245 views

The largest value of $k$ for $\Bbb{Z}^{k}$ to be embedded in $\mathcal{GL}(n,\Bbb{Z})$.

Reading my course on group theory, I asked my self the following question : Suppose that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is the largest value of $k$?
0
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0answers
51 views

Homology groups of $SL(2,\mathbb Z)$

I am reading Brown's book "Cohomology of Groups" and I can't solve exercise II.7.1.3.: "It's a classical fact that $SL_2(\mathbb Z) \cong \mathbb Z_6 *_{\mathbb Z_2}\mathbb Z_4.$ Use Mayer-Vietoris ...
0
votes
2answers
34 views

order of dihedral

I am learning abstract algebra, and I don't quite understand the order of the symmetry of dihedral. When you look at a squares, I agree that there will be 8 symmetry. But all the operations have cycle ...
3
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0answers
39 views

an elementary problem on wreath product groups with combinatorial flavor

Embarrassingly, I got stuck in solving the following elementary exercise. Let $G=H\wr \Gamma$ be a wreath product groups, $H,\Gamma$ are countable discrete groups, when $\xi\in\oplus_{\Gamma}H$, then ...
0
votes
3answers
28 views

How do you compute the inverse of the following permutation?

g = Row#1 (1 2 3 4 5 6) Row #2( 2 3 1 6 5 4) How do you compute g inverse and what is the identity of g?
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1answer
46 views

The relation between orders in a group

G is a group and N is a normal subgroup of G.what is the relation between the order of $x$ and $x.N$?
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3answers
100 views

Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9: Let $G$ be the set of all $2 \times 2$ matrices $ \left( {\begin{array}{cc} a & b \\ c & d \\ ...
3
votes
2answers
34 views

Prove that the symmetric group on $n$ letters, $S_n$, has order $n!$.

Here's my proof in which I've used another theorem to prove this one. I want you suggest me another proof without using this theorem, please. Proof: By the theorem Cardinality of set of injections, ...
0
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1answer
18 views

Isotropy groups of tetrahedron after identifying its sides

If we identify the 4 sides of a regular tetrahedron in $\mathbb{R}^3$ by letting the group of all isometries of the tetrahedron act on it, what would the resulting space look like? The resulting ...
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0answers
36 views

Frobenius subgroups of Sz(8)

Does the Suzuki group Sz(8) has any Frobenius subgroup except $D_{14}$, $F_{20}$ and $F_{52}$?
4
votes
1answer
116 views

Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
1
vote
1answer
51 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
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vote
1answer
48 views

number of cycles in a permutation

I have given a permutation let 2 3 1 5 4 that is if initially my string is 1 2 3 4 5 the after one permutation it will become 1 2 3 4 5 3 1 2 5 4 that is the number in first position will go ...
4
votes
2answers
50 views

How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
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votes
0answers
36 views

Behavior of groups under extension

We know that an extension of a solvable group by a solvable group is solvable. Similarly we can find other properties of group extensions here Can someone provide a reference to these statements where ...
0
votes
2answers
28 views

multiplication of permutation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, $a=(1\ 3\ 5\ 2)$, $b=(2\ 5\ 6)$, $c=(1\ 6\ 3\ 4)$, $ab=(1\ 3\ 5\ 6)$, ...
20
votes
3answers
651 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
2
votes
0answers
31 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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0answers
20 views

In an infinite cyclic field of non zero units, characteristic $\neq 2$, can an element $-u \neq u$ be expressed as $u^t$ for some finite integer $t$?

For the sake of a proof using contradiction ( to be used somewhere), Lets assume that an infinite cyclic field $F$ of non zero units exists with characteristic $\neq 2$ . In this infinite cyclic field ...
4
votes
1answer
92 views

groups of order $p^5$ and exponent p

We know that a group of order $p^5$ is an extra special group. How we can show it doesn't have any abelian subgroup of order $p^4$? Also what is the presentation of this group if its exponent is equal ...
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2answers
26 views

Normal proper subgroup of product of finite simple groups is isomorphic to one of them

I was wondering if anyone can give me a hint/sketch for the following problem, if possible using elementary group theory methods (I am familiar with the material of, say, chapters 1-4 in Rotman). Let ...
1
vote
3answers
49 views

Does $Kg=K$ implies that $g=e$?

$K$ is a subgroup of the group $G$ and for a element $g\in{G}$,does $Kg=K$ implies that $g=e$? where $e$ is the identity element of $G$.
3
votes
2answers
47 views

Intersection of an arbitrary subgroup with one of finite index

I want to show that if $G$ is a group and $[G:H]$ is finite, then so is $[K:H \cap K]$ for any $K < G$. I think I can do this by showing that $k \in K \implies k(H \cap K) = (kH) \cap K$. Is ...
5
votes
1answer
97 views

Additive non-abelian group?

Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So ...
4
votes
2answers
94 views

Is it possible for two non-isomorphic groups to satisfy the same first-order sentences and be equicardinal?

My question is the same as the title. A proof or a counterexample would be nice.
1
vote
1answer
78 views

Let $a_1,a_2,a_3,…,a_n$ be elements of a group $(G,*)$. Show by induction that $a_1*a_2*a_3…*a_n$ always gives the same answer.

Basically I need to prove that a group $(G,*)$ is associative in the general case. To do this I know I have to use induction to show that no matter where I insert parentheses into the equation ...
4
votes
1answer
97 views

Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
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votes
3answers
64 views

Cosets/ Cyclic group

Let $G=\langle a\rangle$ and $H=\langle a^2\rangle$. Find all the right cosets of $H$ in $G$. Additional info: I understand that a right coset of $H$ in $G$ is of the form $Ha=\{ha:h \in H\}$. But I ...
0
votes
3answers
56 views

How do I prove this isomorphism?

Let $f(x)$ be any injective function, and let $f^n$ denote $f$ composed of itself $n$ times, such that $f^1=f(x)$, $f^2=f(f(x))$, $f^3=f(f(f(x)))$ and so on. Let $f^{-n}$ denote the inverse of $f$ ...
0
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3answers
73 views

A question on Abelian Groups

Prove that every subgroup of an Abelian group is Abelian but the converse is not true. I recently stumbled onto this question , but not able to solve it . Please help me out!
4
votes
1answer
55 views

Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Prove that $G=\{g*x \mid x \in G\}$

Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Prove that $G=\{g*x \mid x \in G\}$ My proof: Let $(G,*)$ be a group and $g$ be a fixed element of $G$. Let $x=g^{-1}*y$, where $y$ is ...
3
votes
0answers
75 views

A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
23
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6answers
1k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...