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

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2
votes
1answer
56 views

If G is finite group with even number of elements and has identity e, there is a in G such that a*a=e

My approach is ; I subtract e from G then G-{e} has odd number of elements. For any element in G-{e}, there must be an inverse of that element in G-{e}. Take any element in G-{e}, say b, If b*b=e, ...
3
votes
0answers
40 views

From semidirect to direct product of groups

I was studying some properties of semidirect products, and I noticed this sentence on Wiki Suppose $G$ is a semidirect product of the normal subgroup $N$ and the subgroup $H$. If $H$ is also ...
0
votes
0answers
28 views

Order of elements under injective homomorphism and isomorphism

Define $G(n)= \{ g \in G: ord(g)=n \}$ Need to show (i): if $f: G \to H$ is an injective homomorphism, then $f(G(n)) \subset H(n)$ for all $n \ge 1$, and (ii): $\lvert G(n) \rvert = \lvert H(n) ...
0
votes
1answer
32 views

Is Rational number under multiplication group?

binary operation is defined on Q such that a*b=ab So, it is just simple multiplication. I found out that this is a group with identity=1 and inverse is reciprocal of element "a" in G. But, my ...
1
vote
1answer
52 views

Determine whether $ \{ a + b \sqrt[3]{2} + c\sqrt[3]{4} \mid a,b,c\in\mathbb Z\}$ is an abelian group

Let $R = \left\{ a + b \sqrt[3]{2} + c\sqrt[3]{4} \mid a,b,c \text{ are integers}\right\}$. Consider $(R; +,*)$, with the usual addition and multiplication Question: Determine whether $(R,+)$ is an ...
0
votes
1answer
24 views

Show that group homomorphism reduces the order of an element

If $f:G \to H$ is a group homomorphism, need to show $ord(g)=n \Rightarrow ord(f(g)) \le n $ for all $g \in G$ I suppose one starts from the definition of homomorphism: $f(g \bullet h)=f(g) \ast ...
2
votes
0answers
38 views

Group theory problems manual

It would be really a worthy contribution if someone please,From the point of view ,of covering all the problems which are based on application of theorems of group theory, recommend a manual of ...
1
vote
1answer
31 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
1answer
24 views

Subgroup generated by $2$ and $7$ in $(\mathbb Z,+)$

In group $(\mathbb Z,+)$ , the subgroup generated by $2$ and $7$ is $\mathbb Z$ $5\mathbb Z$ $9\mathbb Z$ $14\mathbb Z$ In general what's the result for any $n$, $m$ instead of $2$ and $7$. I ...
2
votes
1answer
31 views

Question about cyclic subgroup of a non-abelian group of order $8$.

How is it true that a non-abelian group of order $8$ is guaranteed to have a cyclic subgroup of order $4$? Such a subgroup would be normal since it will have index $2$. Finally, what's a good example ...
2
votes
1answer
58 views

Proof Using Lagrange's Theorem

I am working on a problem in Kurzweil & Stellmacher's introductory finite group theory that looks like this: Let $A, B$, and $C$ be subgroups of the finite group $G$. Prove that if $B \leq A$, ...
2
votes
0answers
31 views

Are the 14 Bravais lattices really distinct?

I have learned that there are 14 distinct Bravais lattices in 3D and any other thought lattice form could be reduced to or expressed in one of these 14 forms. But the primitive unit cell for f.c.c ...
-2
votes
1answer
27 views

Group homomorphism problem [closed]

Let $f:G \to H$ be a group homomorphism such that $\#\ker(f)=n$. Prove that $\#f^{-1}(h)\in\{0,n\}$ for every $h \in H.$
0
votes
0answers
60 views

If G is an FC group and abelian-by-finite, then is G finite-by-abelian? [closed]

Let $G$ be an FC-group, i.e. a group in which every conjugacy class has finite order. If $G$ is abelian-by-finite, show that it is finite-by-abelian. Abelian-by-finite means there exists a ...
-1
votes
0answers
44 views

G be a supersoluble group then some x element of non trivial normal subgroup <x> normal in G?

If group G supersoluble then some element x of non trivial normal subgroup satisfy normal in G
4
votes
3answers
54 views

if $\rho: H \to \text{GL}_n({\bf C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful

How do I show that if $\rho: H \to \text{GL}_n(\mathbb{C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful?
3
votes
2answers
50 views

Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?

Let $G$ be a nonabelian group with center $Z(G)$. Let $\rho: Z(G) \to \text{GL}_n({\bf C})$ be an irreducible representation. Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not?
2
votes
1answer
35 views

Check whether two conjugate subgroups are still conjugate in some subgroup

Given a group $G$ and $K<G$. Let $H_1,H_2$ be subgroups in $G$ satisfying: \begin{align} &H_1\triangleleft K;\\ &H_2\triangleleft K;\\ &H_1\sim_G H_2.\quad\mbox{(They are ...
0
votes
1answer
34 views

To find the total number of non isomorphic subgroups of $G$ of order $17$

G be a group of order $17$ .$G$ is cyclic and has two subgroups both trivial , but howto find total number of non isomorphic subgroups of $G$ ? Thanks
2
votes
2answers
48 views

Labelling the Vertices of Dodecahedron

Dodecahedron has 20 vertices. I want to label them by $1,2,3,4,5$ with the following rule. The five vertices of each face should have different labels. Q. What ...
1
vote
2answers
34 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 ...
4
votes
2answers
85 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
3
votes
1answer
112 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about ...
0
votes
1answer
50 views

Why “even number of elements in Group” in this question is given?

I am trying to prove one question about group. "If finite group G has identity e and even number of elements, prove that there is "a" (not equal to "e") such that $a*a=e$." I just don't understand ...
3
votes
1answer
105 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
1
vote
3answers
62 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 ...
3
votes
2answers
38 views

Prove that $S= \langle(1,2,3,4) \rangle$ has 3 conjugates in $S_4$

Some things I know: $S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$ $(2,4) \in N_G(S)$ Number of conjugates = $[G: N_G(S)]$ This seems like such a easy question but it made me realised that I do ...
1
vote
2answers
39 views

Law of Exponents for Abelian Groups

Let $a$ and $b$ be elements of an Abelian group and let $n$ be any positive integer. Show that $(ab)^n = a^nb^n$. Is this also true for non-Abelian groups?
0
votes
1answer
17 views

isomorphism complex numbers and $W \times \mathbb{R}$

I have two groups $(W = \{z \in \mathbb{C} \mid |z|=1\}, \cdot)$ and $(\mathbb{R},+)$ and the direct product $(W \times \mathbb{R}, *)$ where $*: (W \times \mathbb{R}) \times W (\times \mathbb{R}) ...
1
vote
1answer
27 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
1answer
16 views

group generated by union of positive reals and complex numbers with modulus one

Starting from the group $(\mathbb{C}_0, \cdot)$, we have subgroups $W = \{z \in \mathbb{C} \mid |z|=1\}$ and $\mathbb{R}_0^+$ (the strictly positive reals). My question is what the group, generated by ...
3
votes
2answers
102 views

Prove that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$

$\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to ...
0
votes
1answer
28 views

Sylow p-subgroups

I need to find the Sylow $p$-subgroups of the alternating group $A_5.$ So I need to find the maximal $p$-subgroups of $A_5.$ First of all, what are the elements of $A_5$? I know they are the even ...
1
vote
1answer
30 views

Show that $ker(\phi)=H$

Ler $R$ be a ring and $G$ be a group, $RG$ be the group ring and $H$ be a normal subgroup of $G$. So if |$H$| is invertible in $R$, then setting $e_H=\frac {1} {|H|} \widehat {H}$ where ...
2
votes
1answer
37 views

Question on simple subgroup $H$ and a normal subgroup $N$, of $G$

This one is a bit strange to me, mainly the third hypothesis. It goes as follows: Given a group (finite) group $G$, and $N, H \leq G$ such that $N$ is normal in $G$, and $H$ is simple ...
-3
votes
2answers
38 views

Question regarding homomorphism in groups [closed]

Let $f:G\to H$ a group homomorphism and $\mbox{ker}f$ contains n elements. Prove that $\mbox{Im}f$ has either $n$ or $0$ elements.
4
votes
1answer
51 views

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$. I've been trying to prove this given the definition of a free group $F$: given group $F$ and subset $X\subseteq F$, $F$ is free ...
6
votes
0answers
50 views

Finding the $H$-orbit of $W$ using either Magma or Gap. [closed]

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that $W$ is a subset of $G$. How I could find the $H$-orbit of $W$ by using either Magma or Gap (where $G$ acts upon $W$ by ...
2
votes
1answer
31 views

Proving the existence of a bijection between $U_{mn}$ and $U_m \times U_n$ where $(m,n)=1$ , there by proving Euler $\phi$ is multiplicative

Without proving before hand that Euler's phi $(\phi)$ function is multiplicative , can we prove that there is a bijection between $U_{mn}$ and $U_m \times U_n$ , for any pair of relatively prime ...
2
votes
1answer
20 views

G-set without fixed point question

This is a qual exam question: Suppose G is a group of order pq with p < q both prime. Prove that if m ≥ q(p-1), then there exists a G-set A with m elements and without fixed points. Is the fixed ...
0
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0answers
38 views

How can I work out if a certain group presentation implies a certain relation?

I thought that maybe it would be possible to answer this question using the concept of a group presentation. Let $x_1,x_2,\ldots,x_k$ be k different elements of a group G and $k\geq4$. If we ...
3
votes
3answers
71 views

Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
3
votes
2answers
47 views

Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
1
vote
3answers
80 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 ...
2
votes
1answer
38 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
2
votes
1answer
43 views

Finite group with unique subgroup of each order.

Let $G$ finite group, and suppose $G$ has unique subgroup of each order (which divides $G$'s order) - Show that $G$ is cyclic. I reduced the problem to sylow subgroups of $G$ (they are all normal), ...
1
vote
1answer
49 views

Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. [duplicate]

Let $G$ be a finite group. Prove G is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. I am stuck with this :( Would appreciate your help.
2
votes
6answers
60 views

Ring homomorphism $\mathbb{Z}/8\mathbb{Z} \to \mathbb{Z}/5\mathbb{Z}$ implies $1=0$

Let $f$ be an ring homomorphism from $R_1$ to $R_2$ and define $f^*$ as the homomorphism from the group of units of $R_1$ to the group of units of $R_2$. Suppose $f^*$ is surjective, the question is ...
0
votes
0answers
42 views

Show that $<(1,2)(3,4),(1,2,3,4,5)> = A_5$? [duplicate]

I'm revising for my Group Theory exam and saw this in a past year paper question, and I'm not sure enumerating all 60 elements of $A_5$ and showing how I can get them using the given generating set is ...
1
vote
0answers
32 views

Symmetric group $S_4$ [duplicate]

Let $G$ be the Symmetric group $S_4.$ Give a representative of each conjugacy class of $G.$ Then calculate the size of each conjugacy class. I have no idea how to do this.