A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Difference between the set of generators and the alphabet of a free group

What do we mean by saying "a semigroup P is presented by generators and relations". Isn't it right only for the free semigroups? If it's right, we can't distinguish some two semigroups if they are ...
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Abstract Mathematics - Group theory and isomorphism

I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises ...
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64 views

order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.

Let $a, b$ be elements of a group $G$. Suppose that a has order $5$ and that $a^3b = ba^3$. I want to show that that $ab = ba$. Here is what I think: We know that we have $a^1, a^2, a^3, a^4, a^5 = ...
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63 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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39 views

when is a finitely generated abelian group finite?

I've been asked to show that a finitely generated abelian group G is finite iff $G/pG = \{0\}$ for some prime number $p$, and to find a group such that that is true for all prime $p$. Not really sure ...
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37 views

Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
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44 views

I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
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23 views

If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic

I will be grateful for your help If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic
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15 views

Intuition of a theorem in Abstract groups

A theorem in Abstract groups Let $N \triangleleft G$ Then, $1\cdot $ If $H \leq G$ with $N \leq then H/N \leq G/N.$ Morever, if $N \leq K \leq G with K/N =H/N$ then ...
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85 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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26 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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34 views

Not a normal subgroup by left and right coset

If $G = \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} a,b\in (\mathbb{R}) : a \neq 0$) and assume G is a group under matrix multpication Assume that K = ($\begin{pmatrix} s & 0 \\ 0 ...
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27 views

Geometric interpretation of length function of a coxeter group

It is about an exercise in Humphrey's Reflection groups and Coxeter groups exercise 1 section 5.6. Let (W,S) be a Coxeter system. It is assumed throughout the chapter that S is finite. Let $\sigma : ...
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25 views

Map of an element in a group to the conjugation by g

Let G be a group and suppose $g \in G$. $\varphi:G\rightarrow Aut\left ( G \right )$ $g \mapsto i_{g}$ is a Homomorphism with image $Inn\left ( G \right )$ where $Inn\left ( G \right )=\left \{ ...
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48 views

Prove that the funtion f: $G\rightarrow G$, defined by $f(x)=x^k$, $x \in G$ is a permutation of $G$

Help me with this exercise, I could not do it :( Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the function f: $G \rightarrow G$, defined by ...
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Why is the group of unit upper triangular matrices solvable?

Let $GL_n(k)$ be the n by n general linear group over $k$, $B_n(k)$ be the subgroup of $GL_n(k)$ consisting of all upper triangular matrices, and $U_n(k)$ be the subgroup of $B_n(k)$ whose diagonal ...
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42 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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57 views

A group specified by Generators and Relations.

I'm confused with some terms in several definitions. Is an alphabet of a free group the same thing as a generator set of any group? If it is right, then by a given alphabet (set of generators) can be ...
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2answers
37 views

Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?
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9 views

Quotient Groups Composition Series Solvable [duplicate]

Show that all the quotient groups in a composition series of a finite solvable group G are cyclic of prime order. I know a polynomial equation is solvable in radicals if and only if its Galois group ...
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1answer
29 views

Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the ...
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56 views

Coset multiplication giving a well defined binary operation

Let G be a group and let H be a normal subgroup of G. Then prove that the rule of coset multiplication $(aH)(bH)$=$(ab)H$ gives a well defined binary operation on the set $G/H=(aH| a \in G)$ Can ...
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Are there $n$ nilpotent groups of order $n$ for some $n>1$?

Denote $nil(n)$ to be the number of nilpotent groups of order $n$. I checked the numbers $1<n\le 10^7$ , such that $n$ is neither divisble by $2^{11}$ nor a seventh power of a prime. None of ...
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43 views

Showing H is a normal subgroup by calculating left and right coset

If $G = \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} a,b\in (\mathbb{R}) : a \neq 0$) and assume G is a group under matrix multpication Prove that H = ($\begin{pmatrix} 1 & t \\ 0 & ...
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21 views

Proving existence of automorphism

Let $G$ be a cyclic group, and $a,b$ two generators of $G$. Prove that exists automorphism $f$ such that $f(a)=b$ and that if $f$ is an automorphism then $f(a)$ generates $b$. $G$ is cyclic, thus ...
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43 views

Quaternion group and dihedral group.

The group $Q\subset GL_2(\mathbb{C})$ is generated by $\langle A,B\rangle$ $$ A= \left( \begin{array}{ccc}0 & 1 \\ -1 & 0\end{array} \right) B= \left( \begin{array}{ccc}0 & i \\ i & ...
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Prove a function to be an automorphism of a $p$-group $G$.

Let $M$ be a maximal subgroup of a $p$-group $G$. For fixed $g\in G\backslash M$ and $z\in Z(G)\cap M$ of order $p$, the map \begin{align*} \alpha : G&\longrightarrow G\\ mg^i ...
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1answer
33 views

p-nilpotency and normality [closed]

Let $G$ be a finite group and $P$ be a sylow $p$-subgroup of $G$ such that $N_G(P)$ is not $p$-nilpotent, Then why $N_G(P)$ is non-normal in $G$?
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infinite cyclic group is isomorphic to the group of integers under addition.

Theorem: Every finite cyclic group is Isomorphic to $\left ( \mathbb{Z},+ \right )$ Proof: We show first that the map $\phi$ is a homomorphism. Then, show that $\phi$ is a bijection. ...
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When is a centerless group characteristic in direct product with $\mathbb{Z}^n$?

Consider an abelian group $A$ and a centerless group $B$. We can construct the direct product $A \times B$ of these groups, and $Z(A \times B) = Z(A) \times Z(B) = A \times 1 \cong A$. Now, the center ...
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commutative diagram of groups in Lang's algebra

In Lang's Algebra (section I.3), he says that we can describe the third isomorphism theorem $\frac{G}{K}/\frac{H}{K}\cong G/H$ by the following commutative diagram. \begin{array}{c} 0 & \to ...
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60 views

Is this group non-abelian

Is the group $G = \langle a,b \mid a^2b^2ab^{-1}, a^3b^4a^{-2}b^{-3}\rangle$ non-abelian? If $G$ is abelian then $G = G_{ab} = \mathbb{Z}_2$. Thus, to show that $G$ is non-abelian it is necessary and ...
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52 views

Show Latin Square is not a group.

If we fix the first two rows in the above figure, then there are many ways to fill in the remaining rows to obtain a Latin square. Show that none of these Latin squares is the multiplication ...
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75 views

First Isomorphism proof

Theorem: First Isomorphism Theorem Let G and H be groups $\varphi :G\rightarrow H$ a Homomorphism Then, $G/\ker(\varphi) \cong \varphi(G)$ via the Isomorphism ...
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The commutator subgroup $G'$ of $G$, is the set of all “long commutators”

By definition, the commutator is the subgroup generated by all commutators, that is $$G' = \hspace{1mm}<\{aba^{-1}b^{-1}|a,b \in G\}>$$ I'd like to prove that also $$X = \{a_1a_2\cdots ...
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42 views

Burnside Lemma and colorings of a $C_{8}$ graph

I'm trying to determine the number of different colorings of the vertices of a cycle $C_{8}$ graph. Suppose I have 10 colors and I suppose I can use every color as much as I want. I consider two ...
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29 views

Normality is not transitive

Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p= \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix} $, let $L=(p)$, $K=L\times ...
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290 views

Prove that $|G|=8$ with the given condition

Given a group $G$, with $a\in G, b\in G$ such that $|a|=|b|=4, a^2=b^2, ba=a^3b, a\neq b$ and $G=\langle a,b\rangle$. Prove that $|G|=8$. I'm not sure how to begin this exercise. I suppose that the ...
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1answer
18 views

Deducing information about the normal subgroups of a finite group $G$ from its finite cyclic homorphic image?

The following example is taken from the book "Contemporary Abstract Algebra" by Joseph A. Gallian, seventh edition, page#210. If $G$ is a group of order $60$ and $G$ has a homomorphic image of order ...
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42 views

Considering isomorphism of groups and their properties

I have to prove following theorem. Let $\phi : G \rightarrow H$ be an isomorphism of two groups. Then following statements are true. $1. \ \ \ \ \phi^{-1} : H \rightarrow G \ \ \text{is an ...
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Kernel and Image of a map

Note: The OP has restored the question to its original form. An edit by someone else inadvertently changed the codomain from $\mathbb Z_{2p}$ (which makes sense) to $\mathbb Z_{2^p}$ (which does not). ...
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Endomorphism Ring - Definition

Let $G$ be an Abelian group. We may consider the group $\big(\operatorname{End}(G), +\big)$. Next we may endow $\operatorname{End}(G)$ with the composition of functions to make it a ring. Anyway, it ...
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Prove $H$ is isomorphic to $D_6$

Let $$H= \left\{\left.\begin{bmatrix} a & x \\ 0 & b \end{bmatrix} \right| a,b \in (\mathbb{Z}/3\mathbb{Z})^{\times}, x \in \mathbb{Z}/3\mathbb{Z} \right\} \leq ...
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Estimating the cardinality of the union of the conjugates of a proper subgroup.

If $G$ is a finite group and $H$ is a nontrivial subgroup of $G$ such that $H^{a}\cap H=\{e\}$ for all $a\in G-H$, where $H^{a}=aHa^{-1}$ and $e$ is the neutral element of $G$, show that ...
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Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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the length of the conjugate class containing $\alpha$ in $S_n$ [duplicate]

Suppose $\alpha$ $\in$ $S_n$ and there are exactly $n_i$ $l_i$-cycles ($i=1,2, ... ,k)$ (containing $1$-cycles) in the cycle decompostion of $\alpha$ ., then the length of the conjugate class ...
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$ord(f(a)) \leqslant ord(a) $ for every a in G

Let $f: G \to H$ be a group homomorphism. I want to show that: (i) $ord(f(a)) \leqslant ord(a) $ for every a in G (ii) If $ord(a)$ is finite, then $ord(f(a))$ divides $ord(a)$ (ii) If $f$ is an ...
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52 views

Using sylow's theorem to show a group is non-simple

problem Prove any group G, such that |G|=48 is not simple. My solution: $|G|=48=2^4*3=3*16$ so we can choose the prime number to be 2 or 3. $n_2 | 3$ and $n_2 = 1 mod(2)$ ==> $n_2=1$ or $n_2=3$ ...
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44 views

Is it possible to make this proof two-way?

Show that if $H_1$ and $H_2$ are cyclic groups, their direct product is cyclic if their orders are coprime. If cyclic, then there exists an element $(E_1 | E_2)$ such that $(E_1 | ...
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If $G/Z(G)$ is cyclic, why is $G$ only abelian and not also cyclic?

If the factor group with respect to the center of $G$ is cyclic, then $(aZ(G))^n=gZ(G)$ for some $n$ and any $g$, where both $a$ and $g$ are from $G$ (and $a^n$ is, too). Because of the definition of ...