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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center of a group 4

Sorry for the simplicity of the question, but about which conditions we will have $Z(G)K/K=Z(G/K)$ where $1\not=K\trianglelefteq G$ and $G$ is not abelian? I know that is always worth $Z(G)K/K\leq ...
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1answer
115 views

The set that a group can act on it as a permutation group

What can we say about the set that a group can act on it as a permutation group? My question is about the structure of elements of a group G when acts on an arbitrary set Ω as a permutation group. For ...
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5answers
364 views

If $ H $ is a normal subgroup of $ G $, is $ G/H \times H \cong G $?

If $ H $ is a normal subgroup of $ G $, is $ G/H \times H \cong G $? For example, I think $ \mathbb{Z}/2 \mathbb{Z} \times 2 \mathbb{Z} \cong \mathbb{Z} $. I would construct a map as follows: ...
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3answers
1k views

Show that all abelian groups of order 21 and 35 are cyclic.

Show that all abelian groups of order 21 and 35 are cyclic. I have no idea on how to start. Can anyone give some hints?
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4answers
131 views

Let $H \leq G$: prove that $K=\{x\in G:\exists n \in\Bbb{N},\,x^n\in H\}$ is a subgroup of $G$?

$H$ is a subgroup of an abelian group $G$. We define $$K = \{ x \in G : \exists n \in \Bbb{N}, x^n \in H\}.$$ How to prove that $K$ is a subgroup of $G$ ?
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6answers
533 views

How to prove that if group $G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup? [duplicate]

$G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup? I know to prove that a subset $H$ of group $G$ be a subgroup, one needs to (i) prove $\forall x,y \in H:x \circ y \in H$ and (ii) ...
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82 views

Extension of a binary operation

Could you help me with the following problem? Let $X$ be a set with a (not necessarily associative) binary operation. Let $o$ be an element such that $o \not \in X$ and let $X^o=X \cup \{o\}$. Show ...
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122 views

Show that $G$ with matrix multiplication as operation, is a group.

Let $\mathbb{N}_0$ denotes the set of nonnegative integers and $$ S=\left\{ \sum_{i=0}^k a_i 2^i:k \in \mathbb{N}_0,a_i \in \mathbb{Z}\,\,\forall i \right\} $$ Define $$ G=\left\{ \pmatrix {2^a ...
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3answers
51 views

Suppose $m \in \mathbb{Z}^+$ and $H=(\mathbb{Z}_m,+)$. Show that $nH=H$ if and only if $m$ and $n$ are coprime.

Suppose $m \in \mathbb{Z}^+$ and $H=(\mathbb{Z}_m,+)$. Show that $nH=H$ if and only if $m$ and $n$ are coprime. I have no idea what is $H^n$. Can anyone guide me ? Remarks; Guys, sorry for the ...
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93 views

Flow of D.E what is the idea behind conjugacy?

I got some kinda flow issue, ya know? well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the ...
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6answers
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Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
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7answers
162 views

If $g$ is an element in an abelian group $G$ and $H\leqslant G$, must there exist an $n$ such that $g^n\in H$?

Let $G$ be an abelian group and $H$ a subgroup of $G$. For each $g \in G$, does there always exist an integer $n$ such that $g^{n} \in H$?
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182 views

Show that these groups are not cyclic and hence have no such generator?

I think that the groups $\mathbb{Z}^2, C_2\times C_4$ and $\langle(12)(34)(56),(145)(236)\rangle $ are not cyclic and hence do not have such a generator, but not sure how to prove/show this? Help ...
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107 views

$\pi$-radical of group

Let $G$ be abelian and periodic. Let $\mathbb{P}$ be the set of prime numbers, $\pi \subseteq \Bbb P$ and $\pi ^{\prime }=\Bbb P\setminus\pi $. Let $O_{\pi }\left( G\right) =\left\langle ...
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1answer
136 views

Group, normality, and isomorphism question.

Let $A$ be a subgroup of $B$ and $B$ be a subgroup of $G$. $A$ and $B$ are normal in $G$. Show that the image $B/A$ of $B$ in $G/A$ is normal. Show that $G/B$ is isomorphic to $(G/A)/(B/A)$. ...
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1answer
107 views

Conjugate of an abelian maximal subgroup is maximal

Perhaps this is a trivial question. Let $G$ be a finite group, and $M$ be a maximal (proper) subgroup of $G$. Suppose also that $M$ is abelian. How could I prove that if $x\in G$, then $xMx^{-1}$ is a ...
2
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1answer
321 views

Matrix Group induction proof and order of elements question

$H$ is the set of the matrices $A$ of the form $$A= \begin{pmatrix} \cos(\theta) & \sin (\theta) \\ -\sin (\theta) & \cos (\theta)\end{pmatrix}$$ where $0\leq \theta < 2\pi$ is a group ...
3
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1answer
242 views

Symmetries of a Square $U_{24}$ and $\mathbb Z_{8}$ proving or disproving an Isomorphism

The question is as follows. Consider the following three groups: $\mathbb Z_{8}$, $U_{24}$ and the group of symmetries of the square. For each pair, determine if the two groups are isomorphic. If ...
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3answers
200 views

Cayley's Theorem (Simple)

I don't really like to ask questions where i don't understand whats going on at all, but i just can seem to wrap my head around Cayley's Theorem, we went over it in class and i also watched a YouTube ...
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1answer
145 views

Automorphism proof ( simple)

So the problem is given by Let G be a group and define $\pi $ : $\rm\:G \to G\:$ by $\pi(a)$ = $a^{-1}$, for every a in G. Prove that $\pi $ is an automorphism of G if and only if G is abelian. ...
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216 views

Elementary Symmetric group Question

The question is as follows TRUE or FALSE: For n > 1, any element of $S_{n}$ has order less than or equal to n. If true, give a short proof and if false, give an explicit counter-example. I feel a ...
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Problem involving permutation matrices from Michael Artin's book.

Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is $$ P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 ...
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What can be said about two groups with isomorphic derived factors?

The third isomorphism theorem states that we can relate an isomorphic relation between two normal subgroups of a group $G$. My question is can we infer anything about the two groups structures itself ...
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329 views

Order of matrices in $GL_2(\mathbb{Z})$

Let $A\in GL_2\left(\mathbb{Z}\right)$, the group of invertible matrices with integer coefficients, and denote by $\omega(A)$ the order of $A$. How we prove that $$\left\{\omega(A);A\in ...
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Is this group abelian?

I tried to show that the following group is abelian by manipulation the relations but they didn't work. Please show me the right way. The group is $$G:=\left<x,y \mid xyxy^2=yxyx^2=1\right>$$
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1answer
101 views

Group Theory. Transitivity and Normal subgroups.

I would like to show that, If $G$ acts transitively on a set $X$, and $K$ is regular normal subgroup of $G$. Then $G = K \operatorname{Stab}(a)$. ($K \operatorname{Stab}(a)$. w.r.t G) for any $a \in ...
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1answer
84 views

Group extension

I am getting blurred about group extensions. Let $A,B,C$ be groups. If $G=(A{:}B).C$ and $A$ is characteristic in $G$, then $G=A.(B.C)$. But is it also true that $G=A{:}(B.C)$ ? If $G=A.(B{:}C)$, ...
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If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic. [duplicate]

If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is a cyclic group. Let's denote $\mathbb{Z}_n=\langle1_n \rangle$ and $\mathbb{Z}_m=\langle1_m \rangle.$ My proof goes as follows: since ...
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How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. ...
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Normal subgroup of order $p$ is in the center

Prove that in a group of order $p^2$ ($p$ a prime), a normal subgroup of order $p$ lies in the center. (Since this is Exercise 2.9.6 from Herstein's Topics in Algebra, there are some restrictions on ...
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Question on alphabet of a free group and its generating set

A free group (denoted by $F(S)$) on a set $S$, called the alphabet of the free group is set of all concatenations, of the elements of $S$ and its inverses (called alphabets). In some places, I have ...
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2answers
100 views

Group, normality, subgroup question. [duplicate]

Let $G$ be a group, and $N,H$ be subgroups of $G$ with $N$ normal. Show that $$HN = \{hn \mid h \in H, n \in N \}$$ is a subgroup of $G$. Thanks in advance!
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What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?

Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me). How ...
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1answer
119 views

Determine the orbits of isometries of the plane on unordered pairs of points

"Let the group M2 of isometries of the Euclidean plane act on the set S consisting of pairs of unordered points of the plane. Determine the orbits of this action, and for each orbit, the stabilizer ...
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Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
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776 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
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1answer
262 views

Group actions transitive on certain subsets

Let $G$ be a group acting on a finite set $X$. This also gives an action of $G$ on the subsets of $X$ of any given size, and we can ask whether this action is transitive for some specified size of ...
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2answers
61 views

Needed help to show these are isomorphic

How can we show that these two groups are isomorphic, $\mathbb Q^{\times}$ and $\mathbb Q_{\text{positive}}^{\times}\times\mathbb Z_2$. I don't know what to do with $\mathbb Z_2$. thank you very much
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1answer
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$G$ a group, $H \subseteq G$, $xy^{-1} \in H \Rightarrow$ $H \leq G$

This was a question an exam I had the other day that I didn't quite finish. It states that $G$ is a group and $H$ is a non-empty subset of $G$ with the property that for any $x,y \in H$ we have that ...
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References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
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Why are finite Coxeter groups always tame, why are infinite tame Coxeter groups integral

According to Lusztig's Hecke Algebras with Unequal Parameters (also available as arXiv preprint math/0208154v1): Coxeter group. Let $S$ be a finite set, and for any $s,s' \in S$, $m_{s,s'}$ be a ...
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559 views

Generators of Symmetric and Alternating Group

Consider the symmetric and alternating groups $S_n$ and $A_n$ ($n>2$). 1. Does arbitrary $2$-cycle and an arbitrary $n$-cycle in $S_n$ generates $S_n$? 2. If $n$ is odd, does an arbitrary ...
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0answers
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Fundamental group of $Spin^c(2)$?

Is the fundamental group of $Spin^c(2)$, the second complex spin group, also $\mathbb{Z}$? If so, how does one see this? Just to avoid any confusion, my definition is: $$Spin^c(2) = (SO(2) \times ...
5
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2answers
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Finding the 'cube roots' of a permutation

How do I find three elements $\sigma \in S_9$ such that $\sigma^3=(157)(283)(469)$? Since the three $3$-cycles in $\sigma^3$ are disjoint, $|\sigma^3|=\operatorname{lcm}(3,3,3)=3$. Then since ...
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There exists only two groups of order $p^2$ up to isomoprhism.

I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is ...
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What are the finite subgroups of $K^*$ if $K\in \{\Bbb Q,\Bbb R,\Bbb C\}$?

What are the finite subgroups of $K^*$ if $K\in \{\Bbb Q,\Bbb R,\Bbb C\}$?
3
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4answers
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If $g$ is in $N_G(P)$ then $g\in P$ where $P$ is a $p$-Sylow subgroup

Please help me to solve this problem. Let $P$ is a $p$-Sylow subgroup of the finite group $G$ and $g$ is an element such that $\lvert g \rvert=p^k$ then if $g$ is in $N_G(P)$ then $g\in P$. Where to ...
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1answer
117 views

Third quotient of the lower central series

Let $G = F_n$ be the free group with $n$ generators, $x_1,...,x_n$. Let $[a,b] = a^{-1}b^{-1}ab$, and $G_i$ be the $i^{th}$ group in the lower central series. We know that $G_2/G_3$ is generated by ...
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1answer
93 views

Question about quotient groups of abelian groups

Let $G$ be an abelian group and $H$ a subgroup of $G$, if for some $n$, $H/nH = 0$ and $([G : H], n) = 1$, then is it true that $G/nG = 0$?
3
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1answer
278 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...