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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Trouble understanding Cosets and Lagrange's Theorem

Let $G$ be a group and $H$ a subgroup of $G$ I think all the elements of $h \in H$ multiplied (on the left or right) by one element $g\in G$ forms a coset. Intuitively I can see that $|H|$ = ...
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3answers
58 views

For groups and orders show $\forall a,b\in G: |ab|=|ba|$ and $\forall G$ with $|G| = p^n$ for $p$ prime has a subgroup of order $p$

1) $\forall a,b\in G: |ab|=|ba|$. Let $|ab|=n$, then $(ab)^n=e$. But $(ab)^n=(ba)^n$, so $|ba|=n.$ 2) Every group $G$ with $|G| = p^n$ for $p$ prime has a subgroup of order $p$. Does this follow ...
2
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2answers
96 views

How to see that $PSU(2)$ is same as $SO(3)$?

Some background: We have an action of $SU(2)$ on the space of traceless Hermitian matrices, $\mathcal{H}$, via conjugation: $$SU(2)\times \mathcal{H}\to \mathcal{H}, \ (U,H)\mapsto UHU^{-1}.$$ The ...
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2answers
49 views

Find a group $G$ and a subgroup $H$ such that $x,x',y,y'$ in $G$, $xH=x'H$ and $yH=y'H$, yet $xyH$ does not equal $x'y'H$

Find a group $G$ and a subgroup $H$ such that $x,x',y,y'$ in $G$, $xH=x'H$ and $yH=y'H$, yet $xyH$ does not equal $x'y'H$. (alternatively find an example of $G$ and $H$ where $(xH)(yH)$ does not equal ...
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2answers
60 views

Normal Form of Elements in Quotient Groups

Let $G=⟨ S\mid R_1⟩$ be a group, where $S$ is the set of generators and $R_1$ is the set of relations. Let $H=⟨S\mid R_1, R_2⟩$ be the quotient group $G$ obtained from $G$ by adding a (possibly ...
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2answers
68 views

Do solvable groups have elementary abelian characteristic subgroups?

I know that the minimal normal subgroup of a solvable group is elementary abelian, and that it is characteristically simple, but it isn't obvious to me that it is characteristic. Perhaps I'm missing ...
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0answers
21 views

Find number of triangle and quadrangles in $AG_2(3)$

$\textbf{Question-}$ Show that in $AG_2(3)$ there are - i) $72$ triangles ii) $54$ quadrangles iii) $4$ triangles in each quadrangle iv) $3$ quadrangles containing a given triangle My try- I know ...
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1answer
53 views

Proving the group of rotational symmetries of a Tetrahedron has no element of order 6

How would one prove that the group of rotational symmetries of a Tetrahedron, $G$, has no element of order 6? I think that there is no element of order 6 and hence no cyclic subgroup of order 6, ...
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3answers
36 views

$\mathbb Z^n$ as a proper quotient of $\mathbb Z^m$

This question is the successor of this one. Assume that the group $\mathbb Z^n$ is obtained as the quotient $\mathbb Z^m/H$ of $\mathbb Z^m$ for a non-trivial subgroup $H$ of $\mathbb Z^m$. That is, ...
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1answer
53 views

Identifying groups with subgroups isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$

I was playing around with semidirect products and tried finding a non abelian semi direct product of $\mathbb{Z}_2\times \mathbb{Z}_2\rtimes \mathbb{Z}_2$. I couldn't find a group that worked, and I ...
4
votes
2answers
63 views

$\mathbb Z^n$ as a quotient of $\mathbb Z^m$

It is quite obvious that if $n\le m$, the group $\mathbb Z^n$ can be obtained as a quotient of $\mathbb Z^m$. But is the converse statement also true? That is, if $\mathbb Z^n$ is a quotient of ...
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1answer
38 views

About strict inequality in Groups Locally Nilpotent

Let $G$ be a locally nilpotent group and $x \in G$. How can I prove that $[G,x] \neq G$? Note that if $G$ is a nilpotent group then this statement is true, because $G'=[G,G]<G$ and $[G,x]\leq G'$. ...
3
votes
2answers
778 views

How many group homomorphisms are there from Zn to Zm?

In looking up this question, I found this site: Physics Forums. In it, someone claims that $f(x) = kx$ is a homomorphism from the group $\mathbb{Z}_{m}$ to $\mathbb{Z}_{n}$ if $m$ divides $kn$. I ...
2
votes
1answer
34 views

Generators of $\text{GL}_{2}(\mathbb{Z})$ group, good reference book?

Does anyone know, where I can find a reference (preferably a book) which says that the general linear group $\text{GL}_{2}(\mathbb{Z})$ is generated by the set $$\left\{\begin{bmatrix} ...
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2answers
39 views

Proving <G\H>=G

Let $H$ be a proper subgroup of $G$. Prove that $\left<G\setminus H\right>=G$. I get stuck when trying to write an element of $H$ in this way. Specifically we must prove: $$\forall h \in H ...
3
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1answer
43 views

second derived subgroup for the group on two generators $a,b$ with $[a,b]=a$

Reading one of the articles of Higman, I encountered the following reasoning obvious to the author: Let $G=\langle a,b\mid [a,b]=a\rangle$. Then the second derived subgroup of $G$ is trivial. I ...
0
votes
2answers
64 views

Sylow's First Theorm in GRE book (typo?)

In my Gre book, the Sylow's First Theorem is stated as Let $G$ be a finite group of order $n$, and let $n= p^k m$, where $p$ is a prime that does not divide $m$. Then $G$ has at least one ...
3
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2answers
86 views

Why the space S1 and S1/Z_2 is topologically identical?

I am a physicist studying liquid crystals. My research is bit related to topology but I don't have much knowledge of it. Recently I read from a the book Soft matter physics: An introduction that ...
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0answers
60 views

Subgroups of $A_4$

So I have to show the 12 elements and arrange them into their cyclic subgroups. (I know there are $7!/2$, but I just have to show the cyclic subgroups.) That is my question. So I have: $e, $ ...
3
votes
1answer
47 views

Quaternion^2 in the symmetric group

Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is ...
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2answers
51 views

Existence of homomorphism between two groups

Can there exist an onto group homomorphism from $S_5$ to $ S_4$ or from $S_5$ to $\mathbb Z_5$?Is it possible to write the homomorphism explicitly?
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1answer
38 views

Fundamental domains of Dihedral groups

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?
4
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79 views

Outer automorphisms of direct product of finitely generated groups

Let $A,B$ be finitely generated residually finite groups such that $\text{Out}(A)$ and $\text{Out}(B)$ are residually finite. Is $\text{Out}(A \times B)$ residually finite?
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1answer
53 views

Rank of abelian p-group

Let $G$ be an abelian p-group such that $$ G=A_1\times \cdots A_n $$ is the direct product of $n$ cyclic groups then rank$(G)=n$. The minimal number of generator of a group is the rank of G. This ...
2
votes
1answer
42 views

Show $G$ is a group, when there doesn't seem to be an inverse?

I would like to show that $G=\left\{\begin{bmatrix} a&a\\a&a \end{bmatrix}\mid a\in\mathbb{R}\setminus\{0\}\right\}$ is a group under matrix multiplication. I've already verified that ...
1
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1answer
58 views

Epimorphism from G to Z

I've got a problem with this exercise, I'd be thankful if someone could help. Let $G$ be a group and let $f$ be an epimorphism from $G$ to $\mathbb{Z}$. Show that for every positive integer $n$, $G$ ...
2
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0answers
59 views

s4≀s2 visually?

What does the polytope whose symmetry is (exactly, not larger isomorphisms) s4≀s2 look like? Does anyone know of a full decomposition of its construction, i.e. lattices, hasse diagrams, cayley graphs, ...
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4answers
230 views

Combinatorial group theory books

I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free ...
2
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2answers
57 views

Let G be a finite cyclic group of order n. If d is a positive divisor of n , prove that x^d = e has exactly d distinct solutions in G

well i know that for a group to be cyclic then there must exist an element in G for example we call it g such that $G = \langle g\rangle$ and so $g^0 = e$ and $g^0 = g^n = e$ hence ...
2
votes
1answer
57 views

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$.

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$. where $\mathbb{Q}^*:=\mathbb{Q}\setminus\{0\}$ My attempt: Well, I know that $\langle ...
3
votes
3answers
57 views

Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
3
votes
0answers
76 views

A group is abelian [duplicate]

Let $G$ be a finite group. For any two elements $a,b \neq e\in G$ there exits an automorphism $\sigma$ such that $\sigma(a)=b$. Prove that $G$ is abelian. Only thing that I could conclude about the ...
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1answer
53 views

To show a finite group G is nilpotent

Let G be a finite group and G' denotes it's commutator.If order of G' is 2.Then show that G is Nilpotent. What I have tried:G/G' is abelian,so it is nilpotent again G' is nilpotent as it's order is ...
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2answers
65 views

Solvable subgroups in $GL(n,F)$

Is it true, that any solvable subgroup $G$ in $GL(n,F)$ is subgroup of upper triangular matrix in some basis?
0
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0answers
57 views

Finding Semi-direct products of Z/3Z and Z/7Z

This problem originated from trying to find the group isomorphisms of groups of order 21. I already worked out that there's one subgroup of order 7 that's normal, and one subgroup of order 3. Even ...
3
votes
1answer
29 views

Order of groups and elements

(related to this question: Finite Group and normal Subgroup) Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ We define the set $X:= \{g\in G | g^d=1\}$ ...
4
votes
1answer
324 views

Prove that the kernel of a group homomorphism $\phi$ is a subgroup and that $\phi$ is injective

I am solving the following exercise: Let $\phi : G_1 \rightarrow G_2$ be a homomorphism (where $G_1$ and $G_2$ are groups) and $\ker \phi := \{ g \in G_1 \mid \phi(g) = e \}$ now I have to ...
2
votes
1answer
71 views

Finite Group and normal Subgroup

Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ and define the set $X:= \{g\in G | g^d=1\}$. Show: if $H$ is a normal subgroup of $G$ with order d then ...
6
votes
1answer
125 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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2answers
53 views

Hint to prove this $|A/(A\cap B)|\le|G/ B|$

Let $A$ and $B$ are subgroups of $G$. I need hint to show that $|A/(A\cap B)|\le|G/ B|$.
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3answers
220 views

Find order of group given by generators and relations

Let $G$ be the group defined by these relations on the generators $a$ and $b$: $\langle a, b; a^5, b^4, ab=ba^{-1}\rangle$. I need hints how to find order of $G$.
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1answer
30 views

$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
2
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2answers
57 views

Are these two equivalent interpretations of “The only homomorphic images of $G$ are $1$ and $G$”?

Let $G$ be a group, say nontrivial. Aluffi interprets the property "The only homomorphic images of $G$ are $1$ and $G$ [paraphrased]" ($\dagger$) to mean "If there exists a surjective ...
0
votes
2answers
43 views

Homomorphism $\phi: \mathbb{Z} \times \mathbb{Z} \rightarrow G$ proof

Let $G$ be a group. Let $h,k \in G$ and let $\phi:\mathbb{Z}\times \mathbb{Z}\rightarrow G$ be defined by $\phi(m,n)=h^mk^n$. Give a necessary and sufficient condition, involving $h$ and $k$, for ...
2
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2answers
58 views

Product of finite order elements in a group

Let $G$ be a group. Let $a,b\in G$ be of finite order. Prove or disprove: (1) If $ab$ has finite order, then $ba$ has finite order. (2) If $ab$ has finite order, then $a^{-1}b^{-1}$ has finite ...
4
votes
2answers
142 views

Commutator of a group

A commutator in a group $G$ is an element of the form $ghg^{-1}h^{-1}$ for some $g,h\in G$. Let $G$ be a group and $H\leq G$ a subgroup that contains every commutator. $(a)$ Prove that $H$ is a ...
0
votes
1answer
71 views

A monoid with left identity and right inverses need not be a group [duplicate]

Let $G$ be a set with an operation $\ast:G\times G \rightarrow G$ such that: (1) For all $a,b,c \in G$ we have $(a\ast b)\ast c=a\ast (b\ast c)$ (associativity), (2) There is $e \in G$ such that ...
2
votes
3answers
1k views

Calculating the Order of An Element in A Group

First of all, I am very new to group theory. The order of an element $g$ of a group $G$ is the smallest positive integer $n: g^n=e$, the identity element. I understand how to find the order of an ...
1
vote
3answers
44 views

$V_4\triangleleft S_4$

Let $V_4:=\{(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3),\iota\} \leq S_4$. It is possible to show $V_4\triangleleft S_4$ by considering conjugation. However, after long thought on the matter, I don't ...
3
votes
1answer
80 views

Show that the order of $G$ cannot be divisible by $7$.

Show that the order of a proper primitive group of degree $19$ cannot be divisible by $7$. It means as following- This means that a group $G$ which acts on a set $\Omega$ of cardinality $19$ (this ...