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

learn more… | top users | synonyms (2)

1
vote
2answers
21 views

isomorphism inverse with identity.

Let $\phi\colon G\to H$ be an isomorphism. Show that there exists a homomorphism $\phi^{-1}\colon H\to G$ so that $\phi\circ\phi^{-1}=id_H$ and $\phi^{-1}\circ\phi=id_G$. Because $\phi$ is an ...
1
vote
1answer
21 views

Homomorphism of $\langle a,b \mid a^5b^{-3}, b^3(ab)^{-2} \rangle$ and $A_5$

I need to prove that the map from $\langle a,b \mid a^5b^{-3}, b^3(ab)^{-2} \rangle$ to the alternating group $A_5$ defined by $a \rightarrow (12453)$ and $ b \rightarrow (234)$ is a homomorphism. I ...
1
vote
1answer
99 views

Counting theorem problem.

I have a pentagonal shape: I have to find how many different figures can be made, if the star is regarded the same upon rotation and reflection such that each piece can be black or blue. by each ...
1
vote
2answers
47 views

.Show that $H$ is abelian [duplicate]

Let $G$ be a group of $2n$ .Let H be a subgroup of $G$ consisting of only those elements of $G$ which are of order $\neq 2$.Suppose $o(H)=n$ Show that $n$ is odd and $H$ is abelian. Pairing the ...
1
vote
2answers
47 views

isomorphism argument

I am basically wondering why the two groups I have marked in red are isomorphic. I will explain something after the picture: Let's assume that we accept that $G\simeq Z\times Z \times Z\times Z$ as ...
1
vote
2answers
35 views

Soluble-by-finite group

Could someone give me the definition of "soluble-by-finite group"? I have searched in several books, but they always refer to this type of groups without giving a clear definition. Thanks in advance. ...
1
vote
2answers
35 views

general linear group and special orthogonal group

I have this exercise, I only need help with d. a. Show that $SO_2(\mathbb{R})=\{R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}| \theta \in \mathbb{R} \}$ ...
1
vote
1answer
94 views

Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order 7, then $H$ is a normal subgroup of $G.$

(1) Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order $7$, then $H$ is a normal subgroup of $G.$ (2) Suppose that G is a group with $|G|=35.$ Prove that if $G$ ...
1
vote
1answer
41 views

If $S$ is an Infinite Set, and $f(S)\subseteq{G}$ is a Set that Generates a Group $G$, then the Cardinality of $G$ is $\le$ Cardinality of $S$?

As the header says, if we map an infinite set $S$ into $G$ by $f$, such that $f(S)$ generates $G$, then is $|G|\le|S|$? I know that this should be true in the case where $S$ does not include into $G$. ...
1
vote
1answer
58 views

When $\overline{H} \cap \overline{K} = \overline{H \cap K}$?

Let $G$ be a group, $N \trianglelefteq G$ and $H, K \leq G$. When we have $\overline{H} \cap \overline{K} = \overline{H \cap K}$ (where $\overline{X}$ means the image of $X$ with natural map in $G / ...
1
vote
3answers
53 views

$G$ is a commutative group of order 72, which is a product of cyclic groups. What is max order of element?

I'm trying to understand the following practice question that has the given answer. Can someone help? Here are some specific questions: I presume the notation $(\mathbf{Z}/2)$ refers to some cyclic ...
1
vote
1answer
39 views

Center of an abelian group

Prove if $G$ is non abelian group, then exists an abelian subgroup $H$ which contains $Z(G)$ and $H≠Z(G)$.
1
vote
1answer
27 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
vote
2answers
50 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 ...
1
vote
2answers
36 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 $(GL_{2}(\mathbb{C}),\cdot)$. ...
1
vote
1answer
33 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.
1
vote
1answer
36 views

“Length” of an element in a free group

Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the ...
1
vote
1answer
23 views

How can I find the subgroups of $D_5$

So I've found all the cyclic subgroups: $\langle e\rangle$, $\langle r\rangle$, $\langle sr^n\rangle$, and $D_5$ itself (which are 8 subgroups), but how do I know if these are all? How can I find the ...
1
vote
1answer
53 views

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$ Could someone explain/define the multiplication here for me so that I may attempt this problem. Thank you ...
1
vote
2answers
50 views

Prove rk$B$ $\le$ rk$A$ where A and B are free, abelian and finitely generated groups.

Let $A$ and $B$ be free abelian, finitely generated groups. Let $f:A \to B$ be an epimorphism. Prove rk$B$ $\le$ rk$A$. I could really use a verification. That is a question from my exam today. ...
1
vote
1answer
42 views

Is there any group in which number of the normal subgroups is equal to number of the conjugacy classes?

Let $G$ be a group s.t. $|G|\geq 3$. Is there any example of $G$ such that number of the normal subgroups is equal to number of the conjugacy classes?
1
vote
3answers
57 views

Infinite group with finite order elements [closed]

Can you give me an example of an infinite group in which every element has order $3$ (except identity) ?
1
vote
1answer
57 views

Prove that the group algebras $\mathbb{C}Q_8$ and $\mathbb{C}D_4$ are isomorphic. [closed]

I need to prove that group algebras $\mathbb{C}Q_8$ and $\mathbb{C}D_4$ are isomorphic. How can I do this?
1
vote
1answer
55 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
1
vote
1answer
34 views

In general, what ways are there to show if 2 groups are isomorphic?

I take it that if the number of elements of a given order n is not the same between 2 groups, then they are definitely not isomorphic. So for example if I need to show that $C_{25}$ is not isomorphic ...
1
vote
2answers
37 views

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$.

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$. Let $\phi $ be a homomorphism.Then $\dfrac{\mathbb Z_5 }{kerf}\cong Im f$.Now $Im f$ is a subgroup of $S_5$ .Since $kerf $ is a subgroup ...
1
vote
3answers
70 views

is complex number under absolute value a group?

I have just started going over abstract algebra. One of the question is $*$ is defined on $\mathbb C$ such that $a*b=|ab|$ I tried to check three axioms : 1) Associativity 2) identity 3) inverse ...
1
vote
1answer
35 views

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
1
vote
3answers
112 views

Prove that the following set is a group

Prove that that the following is or is not a group. (a) The set S = $\mathbb{R}$ \ {0} with operation defined by a * b = 2ab for all a and b in S. (On the right side of the equation, the operations ...
1
vote
1answer
30 views

How to determine if the subset $K=\{ g\in S_4|2^g=2\}$ is a subgroup of $S_4$?

I'm not sure exactly how to start. I know that the group $S_4$ is a finite set of 4 symbols whose elements are all the permutations of the 4 symbols which sums up to 24 permutations. That is, $S_4$ ...
1
vote
1answer
50 views

Discrete quotient group

I have a hard time understanding quotient groups. For example, I need to make sense of the expression $$\mathcal{S}_3 (1,3,5) / \mathcal{Z}_2 (3,5).$$ Here, $\mathcal{S}_3 (1,3,5)$ is a symmetric ...
1
vote
2answers
41 views

How do I prove $N(aHa^{-1})= aN(H)a^{-1}$ [closed]

Here $a \in G$ and $H$ is a subgroup of $G$. I have no idea where to begin with this. Please help?
1
vote
1answer
52 views

Operation with normal subgroup

I am working on a problem on finite group theory, and would like asking a question on the correct operation of normal subgroup. Suppose that $H$ is normal subgroup of $G$ and the factor group $G/H$ ...
1
vote
1answer
44 views

Characteristics subgroups, normal subgroups and the commutator.

Let $G$ be a group and let $N$ be a normal subgroup of $G.$ Let $N'$ denote the commutator of $N.$ Prove that $N'$ is a normal subgroup of $G.$ What I do know is that the commutator subgroup is ...
1
vote
1answer
32 views

Does N/H=K/H under some terms mean N=K?

Let $G$ be a group and let $K$ be a normal subgroup of $G.$ Now let $H$ and $N$ be normal subgroups of $G$ containing $K.$ Given $N/K=H/K$ can I show $N=H$ necessarily? Is there a way?
1
vote
2answers
122 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
1
vote
3answers
57 views

subgroup is saturated iff it is a direct summand

Let $A$ be a free abelian group of finite rank. Call a subgroup $B \le A$ saturated if for all $a \in A$ and positive integers $n$ such that $na \in B$, the element $a$ belongs to $B$. Call a subgroup ...
1
vote
1answer
46 views

Finitely generated abelian groups isomorphism

Got this on a home assignment and I don't have a clue... How do I determine if $\mathbb{Z}_{12}\times\mathbb{Z}_{18}$ and $\mathbb{Z}_{6}\times\mathbb{Z}_{36}$ are isomorphic? Any hints will be very ...
1
vote
1answer
34 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
1
vote
2answers
41 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
1
vote
1answer
59 views

Number of group actions [closed]

In how many ways can the group $\mathbb{Z}_5$ act on $\{1,2,3,4,5\}$.
1
vote
2answers
93 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
1
vote
1answer
83 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
1
vote
2answers
37 views

Statement of Sylow's Fourth Theorem (single conjugacy class)

I am confused by the statement of Sylow's Fourth Theorem: Let $G$ be a finite group, $p$ a prime. The Sylow $p$-subgroups of $G$ form a single conjugacy class of subgroups. In particular, I do not ...
1
vote
2answers
41 views

order of group generated by two element with some relation.

The group defined by generators $a,b$ and relations $a^{8}=b^{2}a^{4}=ab^{-1}ab=e$ has order at most 16. How to prove that? I have no idea.
1
vote
1answer
40 views

Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$

I am writing another exam in Algebra this week and this time the main topic is automorphism. I was again going through the example exercises and exams from previous years and this problem is giving me ...
1
vote
1answer
39 views

Direct Product of Cyclic Groups and Quotient Groups

Let G = $Z_4$ x $Z_6$ be the direct product of cyclic groups $Z_4$ and $Z_6$. Let N = <(2,3)> be a normal subgroup of G. Show that G/N $\simeq$ $Z_{12}$ What I have so far.. Given that |N| = 2, ...
1
vote
2answers
62 views

Find all subgroups of group

Given multiplicative group of integers modulo 13: $Z_{13}^*$. Find all subgroups of this group. I need to proove, that this group is cycled. Also, as $|Z| = 12$, I know, that if $H$ is a subgroup of ...
1
vote
3answers
43 views

let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. 1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$. 2) ...
1
vote
1answer
28 views

The influence of the finiteness of a set on the conjugation classes of a group

Let $G$ be a torsion group. Suppose that $G = \langle a,B \rangle$ where $a \in G$ and $B$ is a abelian subgroup of $G$. Denote by $a^B$ the set of the conjugates of $a$ by elements of $B$, i.e., $a^B ...