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

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Abstract algebra question: abelian group.

$H=\left\{\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix} : b \in\mathbb{R}\right\}$ $G=\left\{\begin{pmatrix}a & b \\ 0 & d\end{pmatrix}: a, b, d \in\mathbb{R}, ad\ne0\right\}$ $H$ is ...
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2answers
30 views

Does conjugation preserve spectrum of matrices?

Actually, I saw normalizer of diagonal matrices are permutation matrices. I read the answer but I don't know how to prove that conjugation preserves the spectrum. Actually I do some proof on 2x2 ...
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3answers
68 views

No non-zero ring is a group under multiplication

How do I prove that every non-zero ring is not a group under multiplication?
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1answer
64 views

When does $(ab)^c = e \; \Rightarrow \;a^c = b^c = e$?

Let $G$ be a group. Suppose that $a, b\in G$ have finite orders $n$ and $m$, respectively, and suppose also that $ab = ba$. Now, given these assumptions, it is clear that $$c\in \mathbb{Z}\;\wedge ...
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1answer
43 views

if $g^k=e$ then $\chi(g)=\sum_j^n \zeta_k$

Let $G$ be a group. Let $g \in G$ and $g^k=e.$ Let $\chi$ be an $n$-dimensional character of the group $G.$ Let $\zeta_k$ be $k$-th root of unity. Prove that $\chi(g)$ is equal to sum of a ...
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3answers
54 views

two question about isomorphism.

Show that the group of rational numbers $\Bbb Q$ doesn't contain non-trivial subgroups $G$ and $H$ such that $\Bbb Q$ is isomorphic to $G \oplus H$ Let $R$ be $\Bbb Z_2[x]/\langle x^2 + 1 \rangle$. ...
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2answers
36 views

Isomorphism class of $\mathbf{U}(p^n)$

Note that $\mathbf{U}(k)$ is the unitary group. i.e. $\mathbf{U}(k)=\{x<k | \gcd(x,k)=1\}$ We need to find the isomorphism class of $\mathbf{U}(p^n)$ where $p$ is an odd prime. The isomorphism ...
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2answers
84 views

What are the good textbooks on Kac-Moody groups?

While there is a number of good books on Kac-Moody algebras ("Infinite dimensional Lie algebras" by Kac is already enough), it seems to me there is lack of textbooks on Kac-Moody groups. nLab says ...
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3answers
128 views

Can we make an abelian group into a ring by defining multiplication only on a generating set?

Suppose we have an abelian group $(G,+)$ that is generated by some set $A\subseteq G$. Suppose that we are able to define a binary operation $\ast$ on $A$, i.e. $$\ast:A\times A\to A,\quad ...
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0answers
39 views

Definition of K-normal subgroups

"We say that a group K acts semisimply on an abelian group A, if the intersection of all maximal K-normal subgroups of A is trivial" from "Construction of Finite Groups" article by "BESCHE and EICK" ...
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1answer
67 views

Painting a cube using N different colors?

In how many different ways can a cube be painted by using N different colors of paint? Note that this question is not same to Painting the faces of a cube with distinct colours as the colours ...
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1answer
56 views

Showing that $|P|$ divides $|A|!$ where $P$ is a $p$-group and $A$ is maximal abelian normal

I'd appreciate verification of the following proof. This is the part following what I asked in this question. Thanks. If $A$ is as in the linked-to question above (that is, $A$ is maximal among ...
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1answer
32 views

Coset of the Abstract index group of a Banach Algebra?

I'm studying on the book of Douglas: "Banach algebra techniques in operator theory" and there is a passage I don't understand, and I hope you can give me a hand. "A continuous function $f$ from $X$ ...
2
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2answers
97 views

Product of disjoint cycle

I've found the question to find product cycle of let be $\phi$ = (2, 4, 9, 7,) (6, 4, 2, 5, 9) (1, 6) (3, 8, 6) in S9. I know how to express the permutation of S9 as product of disjoint cycle, like ...
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1answer
37 views

How to find the number of transposition

I just learning the abstract algebra now, I'm stuck to find how many transpositions can be made from $(1\ 8)(2)(3\ 6\ 4)(5\ 7)$?
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1answer
45 views

Universal Algebras for Pseudovarities and their cardinality

A Birkhoff variety is a class of algebras closed under division and arbitrary products, a pseudovariety is a class of algebras closed under division and finite products. Now for each type of ...
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1answer
33 views

How to find how many cosets are of $H \cap K$?

I'm confuse to find how many cosets of $H \cap K$ are in the G? If $G$ is a group of order 48, then $H$ of order 8, $K$ of order 6, <= $G$.
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1answer
46 views

Why is the collections of all groups a variety

A variety is an equationally defined class of algebras. As I understand it equationally defined means defined by universally quantified equations, for example the variety of all semigroups could be ...
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1answer
52 views

How to show that $A_n$ generated by all permutations of the form (ab)(cd)?

I need to show that if n $\ge$ 5 then $A_n$ generated by all permutations of the form (ab)(cd) (a b c d are all different) Can I use the fact that union of conjugacy classes is normal ? Its clear ...
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1answer
82 views

Commutator subgroup of a finitely generated nilpotent group.

Let $G$ be a finitely generated nilpotent group. Is the commutator subgroup $[G,G]$ finitely presented? Edit: I am also interested in the weaker question: is $[G,G]$ finitely generated?
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2answers
44 views

Help with a proof envolving a finite group and a specific bijection

Let $G$ be a finite group, and let $k>1$ be an integer. I need to prove that if the mapping $f:G\rightarrow G$, defined by $f(g)=g^k$, is bijection, then $\gcd(k,|G|)=1$. I almost certain that if ...
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1answer
79 views

Group action on set of maps - formula

It is given that $G:X$ and $G:Y$. Does this $[g\bullet f](x) := g\bullet f(g\bullet x)$ formula define group action $G:(Y^{X})$ I guess it doesn't, but I can't prove it as for now. And there must be ...
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1answer
48 views

Subgroup of $S_5$ contains cycle of length $5$ and transposition

Suppose a subgroup of $S_5$ contains a cycle of length $5$ and a transposition. Must it be all of $S_5$? Say, it contains the cycle $(1 2 3 4 5)$ and a transposition $(ij)$. Then it contains a ...
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Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
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1answer
47 views

For $\frac{(G/H)}{(M/H)}$ to be defined, is it necessary that $M\unlhd G$?

Let $M\leq G$. Also, $H\unlhd G$ and $H\unlhd M$. For $\frac{(G/H)}{(M/H)}$ to be defined, is it necessary that $M\unlhd G$? A premilinary investigation shows not, but I can't find a counter-example ...
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4answers
120 views

Why is $\operatorname{Hom}(A, B)$ an abelian group?

Can someone please explain why a Hom-set (the set of all morphisms between two abelian groups $A$ and $B$) does also form an abelian group with addition? By the way both groups $A$ and $B$ have the ...
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1answer
95 views

Find all subgroups of $\mathbb Z_2\times\mathbb Z_4$ - Fraleigh p. 110 Exercise 1.11

I tried to fill in the steps but I'm still confounded by this solution. Any other subgroup must have order 4, since by Lagrange's Theorem, the order of any subgroup must divide 8 and: The subgroup ...
6
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1answer
617 views

Find all proper nontrivial subgroups of Z2 x Z2 x Z2 - Fraleigh p. 110 Exercise 11.10

$\newcommand{\lcm}[0]{\mathrm{lcm}}$I tried to fill in the steps but I'm confounded by this solution. Here $i$ is the identity element, not $e$. Because $\lcm(2, 2, 2) = 2$ hence all non-identity ...
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312 views

Why is a group of order 135 nilpotent?

Why does order 135 imply nilpotent?
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1answer
98 views

Some questions concerning the symmetric group $S_n$

Let $a_n$ be the number of permutations in $S_n$ having an square root. Is it true that $a_{2n+1} = (2n+1)a_{2n}$ ? (experimental data's shows that this is true for small values of $n$). Is there ...
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1answer
27 views

Given the generators of a group find the parametrization matrix

I have the generators of $sl(2,\mathbb{R})$ algebra $$J_0=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\quad J_1=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\quad J_2=\begin{pmatrix}0&1\\ ...
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1answer
31 views

Writing $p$-groups using $p$-adics

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.
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2answers
45 views

Indecomposable group

By denition, an indecomposable group $G$ is a nontrivial group that cannot be expressed as the internal direct product of two proper normal subgroups (i.e : if $H,K< G$ such that $G\cong H\times K$ ...
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2answers
66 views

$G$ is a finite group, Show that if $[G:Z(G)] = 21$ then exists $H ,K < G$ such that $H\neq G$ and $K \neq G$ and $HK = G$

I need to use the correspondence-theorem in order to prove it. $G$ is a finite group, Show that if $[G:Z(G)] = 21$ then exists $H ,K < G$ such that $H\neq G$ and $K \neq G$ and $HK = G$.
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100 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
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4answers
83 views

Homomorphisms from $Z_n$ to $Z_m$

I'm reviewing my Abstract Algebra and I'm stuck on something. My professor explained that if $\varphi:G\rightarrow H$ is a homomorphism, then $$\varphi(1_G)=1_H$$where $1_X$ is the identity in the ...
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3answers
97 views

How many Homomorphisms can be between $Z_6$ to $Z_{18}$?

How many Homomorphisms can be between $Z_6$ to $Z_{18}$? and the most important: What is the algorithm for calculating, step by step?
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1answer
27 views

Questions about definition of edges in affine building.

I have a question about about edges in a building in the book of expander graphs by Alexander Lubotzky, page 69. We know that if $L_1' \subseteq L_2'$ and $[L_2' : L_1'] = p$, then there is an edge ...
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3answers
98 views

(Abstract Algebra) Group Structure Axiom Question

For my homework, I have been asked to analyze the following pair of set and binary operation: N = The set of integers, subtraction. So I'm trying to figure out what the identity is for this ...
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0answers
20 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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5answers
112 views

Is the notation $X²$ an abuse of language in a polynomial ring.

I wonder if the notation $$ P(X) = a_{0} + a_{1}X + a_{2}X²$$ where $X$ is an indeterminate variable is an abuse of notation. Is $X^2$ just $X_2$? Put it another way, what's the meaning of the power ...
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1answer
51 views

Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
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1answer
37 views

Subgroups of $\mathbb{D}_6^n$

Let $\mathbb{D}_6=\{1,x,x^2,y,xy,x^2y\}$ be the Dihedral group of order 6. I'm trying to find two subgroups $N\le \{1,x,x^2\}^n$ and $M\le \{1,y\}^n$ such that $MN=NM$ (so that the product is also a ...
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2answers
27 views

prove that if $k|m$, $\mathbb{Z}_m$ has a subgroup of order $k$ [duplicate]

prove that if $k|m$, then $\mathbb{Z}_m$ has a subgroup of order $k$. im not sure where to start with this. any help is appreciated. thanks!
3
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1answer
41 views

Prove there is no solution of the equation

I need to prove that there is no solution of the following equation $$x^3\equiv 2\pmod{151}$$ I think that it has something to do with "Fermat little theorem" and/or "Euler theorem", but I can't ...
2
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2answers
70 views

Fast way of calculating the order of an element in $\mathbb{Z}_n$?

Is there a fast way of calculating the order of an element in $\mathbb{Z}_n$? If i'm asked to calculate the order of $12 \in \mathbb{Z}_{22}$ I just sit there adding $12$ to itself and seeing if the ...
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1answer
44 views

Unique Complete Reducibility of Finite Groups

Maschke's Theorem states that every complex representation $(\rho,V)$ of a finite group $G$ can be written as a direct sum of irreducible representations that form subsets of V, such that $V = ...
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1answer
178 views

The only element of $S_{\large{n \ge 3}}$ satisfying $\sigma y = y\sigma$ for all $y \in S_n$ is the identity permutation - Fraleigh p. 86 8.47

I don't want to type Greek letters hence I replaced $\gamma$ by $y$. Microsoft didn't replace them all. Call the identity permutation $id$. Prove the contraposition: $\sigma \neq id \implies ...
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1answer
60 views

If $a$ is an element of a group and $|a| = n$ , prove that $ C_G(a)= C_G(a^k)$

If a is an element of a group and |a| = n , prove that $ C_G(a)= C_G(a^k)$ when k is relatively prime to n. $C_G(a)$ refers to centralizer of $a$ Attempt : x $\in$ G and x $\in$ C(a) iff $ a x = xa $ ...
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1answer
54 views

Consequences of Schur's Lemma

Schur's Lemma states that given two irreducible representations $(\rho,V)$ and $(\pi,W)$ of a finite group $G$ (V and W are linear vector spaces over the same field), and a homomorphism $\phi\colon ...