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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85 views

Finitely many group extensions?

Given a group $H$ and a finite group $F$. Consider the group extension $1\to F\to G \to H \to 1$. My question is, are there only finitely many isomorphism classes of $G$ satisfying the above exact ...
5
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1answer
84 views

No Natural Group Structure on Conjugacy Classes

Is it possible to show that there is no natural group structure on conjugacy classes in a group? Alternatively, for a path connected space $ X $, the set $ [S^1,X] $ of free (unpointed) homotopy ...
2
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4answers
70 views

Proof verification that $A_4$ has no subgroup of order $6$.

Assume that $A_4$ has a subgroup of order $6$, and let $H$ be a subgroup of order $6$. Since the order of $A_4$ is $12$ and the order of $H$ is $6$, we know that there are two left cosets. Saying ...
1
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1answer
64 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
0
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1answer
31 views

Not understanding a line in a proof concerning Monomorphism and injectivity

In the proof that "in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective" given in Wikipedia ...
0
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2answers
67 views

Right cancellation of homomorphisms on groups implies

Let $f:G \to H$ be a a group homomorphism such that for any two groups $H_1 , H_2$ , and any homomorphisms $g_1 : H \to H_1 , g_2 :H \to H_2$ , $g_1 \circ f = g_2 \circ f \implies g_1=g_2 $ ; then is ...
4
votes
1answer
99 views

Looking for examples of an uncountable proper subgroup of $(\mathbb R,+)$ without using the concept of Hamel basis of $\mathbb R$ over $\mathbb Q$

Please give some examples of an uncountable proper subgroup of $(\mathbb R,+)$ that does not depend on Hamel basis of $\mathbb R$ over $\mathbb Q$ . Using Hamel basis this is easy as we can find an ...
0
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1answer
40 views

Is there any easier way to find out every proper subgroups of Dihedral group 4?

There are 8 elements in Dihedral group 4 When finding proper subgroups of this, do I have to list every element and draw a table to find out subgroups ?
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2answers
56 views

Topological group closed path

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$, $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let $f,h$ be closed paths ...
0
votes
0answers
17 views

Determine the center of $\text{GL}_n(\mathbb{R})$ [duplicate]

Determine the center of $\text{GL}_n(\mathbb{R})$ Where $\text{GL}_n(\mathbb{R})$ is an algebraic group of all $n \times n$ invertible matrices. The center is the subset of those matrices that are ...
2
votes
1answer
61 views

Group theory, order with permutation of Z

Let $S_{Z}$ be the set of permutations of ${Z}$ (note that this is an infinite group!). Find two elements of $S_Z$ which both have finite order, but whose product has infinite order. I just am really ...
1
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1answer
36 views

Is identity permutation cyclic?

One of the theorem that I have a trouble with is If σ is a cycle of length n, then σ^r is also a cycle if and only if n and r are relatively prime If σ=(123), then σσσ becomes identity permutation. ...
1
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1answer
36 views

homotopic closed paths in topological group

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in ...
0
votes
1answer
53 views

If σ is a cycle of length n, then σ^r is also a cycle if and only if n and r are relatively prime

If σ=(1.2.3), σσσ= identity permutation, which is cyclic in this case n=3 and r=3 but their gcd is not 1. I don't understand why -> this direction of theorem is true
1
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1answer
59 views

Commutator subgroup is normal?

Let $P$ be a group, $H$ a subgroup of $P$, $G$ a normal subgroup of $P$, $[G,H]$ the subgroup of $P$ generated by commutators $[g,h] = ghg^{-1}h^{-1}$, with $g \in G$ and $h \in H$, and $P' = [P,P]$. ...
2
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2answers
57 views

The free group given by $\langle a,b:a^2=b^3=e\rangle$ is not abelian.

Let $G=<a,b:a^2=b^3=e>$. I'm trying to show that $G$ is not abelian. $G$ is by definition given by $F(\{a,b\})/N$ where $F(\{a,b\})$ is the free group on 2 letters and $N$ is the smallest normal ...
0
votes
1answer
45 views

Is H a subgroup of R$^*$?

H = {a + b$\sqrt{3} \in R^*$|a, b $\in Q$}. Is H a subgroup of $R^*$? 1 $\notin$ H so H should not be a subgroup, but the answer is given that it is a subgroup.
0
votes
1answer
25 views

Why is a semigroup $H$ of prime size with proper subgroup $G$ not group?

I know that this is addressed in a corollary of Langrange's theorem, which states that if a group $H$ has a prime order, there exists no nontrivial subgroups, since the cosets of any subgroup must ...
2
votes
2answers
66 views

Finding the character table of $Q_8$

Assume $G$ is a finite group. I am trying to construct the character table of $Q_8$, which is defined by $$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$ By considering the ...
0
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2answers
31 views

Show that group actions are the same as homomorphisms $G \to \text{Perm}(X)$

The following is a homework problem. The conclusions are extremely intuitive and easy to see, but I am having proving this. Could someone please help? Show that, given a group action $G \times X \to ...
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vote
3answers
90 views

If $a^{-1}$ has a cube root, so does $a$.

If $a^{-1}$ has a cube root [in a group], then so does $a$. Attempt Suppose $a^{-1} = bbb.$ Then, $b^{-1}(a^{-1})b^{-1} = b^{-1}(bbb)b^{-1} \to b = b^{-1}(a^{-1})b^{-1}.$ Now I am not sure if it ...
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2answers
42 views

Find the order of this matrix on the group $(GL_{2}(\mathbb{C}),\cdot)$.

I have to calculate the order of the matrix \begin{equation} A= \left( {\begin{array}{cc} i & 0\\ -2i & -i\\ \end{array} } \right) \end{equation} on $(GL_{2}(\mathbb{C}),\cdot)$. ...
0
votes
1answer
24 views

$|D_{18}|=18=2 \times 3^2$, then $D_{18}$ have 2-sylow and 3-sylow subgroups.

Let $G=D_{18}=\langle a , b | a^9=b^2=1 , bab=a^{-1} \rangle$. Then $D_{18}=S_3 \times Z_3$? $|D_{18}|=18=2 \times 3^2$, then $D_{18}$ have 2-sylow and 3-sylow subgroups. 3-sylow subgroup of ...
5
votes
1answer
56 views

A compact infinite topological group with only two closed subgroups

It can be proved every compact infinite abelian topological group $(A,\tau )$, with $\tau$ nontrivial, has at least three distinct closed subgroups. Is there any compact infinite non-abelian ...
1
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1answer
58 views

How many subgroups does $\mathbb{Z}_5$ have

How many subgroups does $\mathbb{Z}_5$ have (addition)? here is my Cayley table (sorry for the formatting): $\ \ \ 0\ 1\ 2\ 3\ 4\ \\ 0\ 0\ 1\ 2\ 3\ 4 \\ 1\ 1\ 2\ 3\ 4\ 0 \\ 2\ 2\ 3\ 4\ 0\ 1 \\ 3\ 3\ ...
4
votes
3answers
156 views

If we have a subgroup of index 3 that is not normal, show there is a subgroup with index 2

Given a a subgroup $H$ of $G$ with index $3$, we have to show there is a subgroup $K$ of $G$ with index $2$, assuming that $H$ is not a normal subgroup of $G$. My line of thinking was the ...
0
votes
1answer
23 views

Question regarding symmetric difference - Please check my work

Let $S$ be a relation on $P(\Bbb{R})$, such that $\displaystyle S=\{<A,B>\in\left(P(\Bbb{R})\right)^2.\left|A\triangle B\right|\le \aleph_0$ Is $S$ an equivalence relation? My try: ...
1
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1answer
50 views

The intersection of subgroup and normal subgroup: the greatest common divisor?

Is the order of intersection of subgroup $H$ and normal subgroup $N$ of group $G$ the greatest common divisor of $\lvert H\rvert$ and $\lvert N\rvert$?
0
votes
1answer
67 views

How many proper nontrivial subgroups do D5 have?

Do I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups
1
vote
3answers
85 views

Proving subgroups

Question: Let $H$ be a subgroup of $G$ and let $K=\{x \in G: xax^{-1} \in H\ \iff\ a \in H\}$. Prove: (a) K is a subgroup of G So for (a): For closure, I need to show: (i) if $a \in H$ then ...
1
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1answer
32 views

Show that the right cancellation law holds in $S$

Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z = y$. (This $z$ may depend on $x$ and ...
2
votes
1answer
60 views

when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, ...
0
votes
1answer
38 views

Number of elements in a group ring

Let $R$ be the group ring $\mathbb{Z}_5 S_3$, where $S_3$ is a symmetric group. I need to calculate the number of elements in the group ring, and I'm not sure how to do it, is it just $5^3 = 125$?
3
votes
0answers
55 views

Prove that $G \times H$ is a group.

$G \times H = \{(x, y): x \in G \text { and } y \in H\}.$ The operation on $G \times H: (x, y) \cdot (x', y') = (xx', yy').$ $(x_1, y_1)[(x_2, y_2)(x_3, y_3)] = (x_1, y_1)(x_2x_3, y_2y_3) = ...
1
vote
0answers
19 views

Automorphism of a group [duplicate]

If Automorphhic group of Group G is trivial group then how will you show that order of group G is less than or equal 2.
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0answers
35 views

Number of the subgroups of a $p$-group with order $p^k$ is congurent to $1$ modulo $p$

Let $G$ be a $p$ group of order $p^n$ and $k\leq n$. Theorem:Number of the subgroups of $G$ with order $p^k$ is congurent to $1$ modulo $p$. I have found a proof of this theorem in Rotman's book ...
0
votes
1answer
60 views

What is the difference between the representation of a group and an algebra?

Sometimes, I come across this idea in physics -> the representation of Lorentz group: SO(3,1) and the representation of Lorentz algebra: so(3,1). At times, I mix them up. Is there a good intuitive way ...
1
vote
2answers
30 views

Elementary question on stabilizer and S3

Let $S_3$ be our group. How can I show that $C_G(x) \text{ for }x=(1\,2\,3)$ is $\{1, (1\,2\,3),(1\,3\,2)\}$ without testing every element in $S_3$?
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2answers
127 views

Prove that any element in $S_n$ can be written as a finite product of the following permutations

Prove that any element in $S_n$ can be written as a finite product of the following permutations: $(a)\ (12),(13), . . . , (1n)$ $(b)\ (12),(23),...,(n−1,n)$ $(c)\ (12),(12\dots n)$. I have no ...
2
votes
2answers
53 views

Let $G$ be an abelian group. Let $\phi$ be a morphism on $G$. Can I say that $\phi(a)\cdot \phi(b)=\phi(b) \cdot \phi(a)$?

My question is this one: Let $G$ be an abelian group. Let $\phi$ be a morphism on $G$. Can I say that $\phi(a)\cdot \phi(b)=\phi(b) \cdot \phi(a)$, with $a$ and $b$ elements of $G$?
2
votes
2answers
53 views

Determine whether or not $H$ is a subgroup of $G$ (assume that the operation of $H$ is the same as that of $G$)

I'm trying to answer the question: Determine whether or not $H$ is a subgroup of $G$ (assume that the operation of $H$ is the same as that of $G$). $$G=\langle\mathbb{R}, +\rangle,\ ...
2
votes
2answers
68 views

Is $A_n$ non-abelian for $n= 3$?

In the book, it is asked to show that $A_n$ is non-abelian for $n ≥ 4$. Which may imply that it is abelian for $n=3$. Is that so? because $(13)(12)\ne (12)(13)$. Hence is it true to write: $A_n$ is ...
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vote
0answers
50 views

Every odd permutation in Sn can be written as a product of 2n+3 transpositions?

My question is If Set of permutation set Sn for n>3 Prove Every odd permutation in Sn can be written as a product of 2n+3 transpositions and every even permutation as a product of 2n+8 ...
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0answers
50 views

How do I visualize algebraic products?

I think I have no or a little problem with analyzing given algebraic products. That is, I know properties of direct, semi-direct and free products. For example, the free product $G\ast H$ is a group ...
-3
votes
1answer
69 views

Show there is no non-abelian group of order 9 [duplicate]

I want to show there is no non-abelian group of order 9. How should I attempt this?
1
vote
2answers
54 views

A cyclic group of order $n$ can be generated by $a^k$ if $(k,n)=1$

I am trying to provide a solution to the following exercise. Please point out anything that you find wrong and/or bad. Show that a cyclic group $G$ of order $n$ generated by an element $a$ can ...
0
votes
1answer
42 views

Let $G$ be a group. Prove that the following affirmations are equivalent.

I'm having problems doing this exercise: Let $G$ be a group. Prove that are equivalent: $(i)$ G is an abelian group. $(ii)$ $\phi(x)=x^{-1}$, $\forall\ x\in G$, is a morphism. ...
10
votes
1answer
121 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
1
vote
1answer
49 views

Prove that the group of rigid motions of a cube contains 24 elements.

How can I prove that the group of rigid motions of a cube contains 24 elements. Thank you.
3
votes
1answer
47 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...