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

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9
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3answers
3k views

Does every group whose order is a power of a prime p contain an element of order p?

I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in ...
2
votes
2answers
157 views

Why is the image of a homomorphism a subgroup of the codomain? [duplicate]

Why is an image of a homomorphism a subgroup of the codomain? I know that the image of any function is a subset of the domain, but I don't know how to proceed with the former statement.
5
votes
1answer
312 views

Subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$

I am trying to show that the subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$, if $n\ge 2$. I tried to do this as follows: Let $G < S_n$ be a subgroup of index $n$, and let it act on ...
3
votes
1answer
112 views

Showing that a homomorphism between groups of units is surjective.

Let $n$ be a positive integer and let $d$ be some divisor of $n$. Consider the group of units modulo $n$, which we shall denote by $U(n)$. Likewise, denote the group of units modulo $d$ by $U(d)$. ...
2
votes
1answer
46 views

Induced injections between free groups

Let $A$ and $B$ be non-empty sets with associated free groups $F(A),F(B)$. Given an injective function $f: A \to B$, is the induced homomorphism $\bar{f}: F(A)\to F(B)$ injective? Let $i_A: A \to ...
-2
votes
3answers
146 views

Are the irrationals + zero an additive group?

Is it true that the only way two irrationals can sum to a rational is if they sum to zero? Thanks! -Dan
2
votes
1answer
258 views

Möbius transformations forming a group and isomorphism with $S_{3}(D_{6})$

My task is to prove that the Möbius transformations defined by $z,\frac{1}{z},1-z,\frac{1}{1-z},\frac{z}{z-1},\frac{z-1}{z}$ make up a group that is isomorphic to the group $S_{3}(D_{6})$. Identify ...
4
votes
1answer
120 views

Minimum size of the generating set of a direct product of symmetric groups

Let $m$ and $n$ be positive integers. Let $S_m$ and $S_n$ be the symmetric groups on the sets $\{1,\dots,m\}$ and $\{1,\dots,n\}$, respectively. What is the minimum size of a generating set for the ...
1
vote
1answer
88 views

$A_n$ contains an isomorphic copy of $S_{n-2}$ [duplicate]

Question is to prove that : $A_n$ contains an isomorphic copy of $S_{n-2}$ I have no idea from where to start. I do not even believe this can be true (expect for cardinality conditions). But as ...
1
vote
1answer
72 views

Understanding a proof about finite $p$-groups

I can't follow the reasoning of the author,in this proof: let $G$ be a finite $p-$group. If $H$ is a proper subgroup of G, then $H<N_G(H)$ (clearly $N_G(H)$ is the normalizer of $H$ and p is ...
2
votes
1answer
332 views

understanding lattice in detailed

I would like to understand meaning of lattice in mathematics, for example let us consider its application, first one is Elliptic function: In complex analysis, an elliptic function is a meromorphic ...
1
vote
2answers
47 views

Is linear space totally different from group?

Linear space builds on Abelian group. My question is, is linear space TOTALLY different from group? Is it true that some properties of linear space are the properties of the Abelian group? Actually, ...
1
vote
0answers
52 views

Product and quotient in Abelian groups

In linear algebra, for any vector space $V$ and its subspace $U$, there exists a subspace $W$ of $V$ such that $U\oplus W=V$ and $W\cong V/U$. Does similar property hold for Abelian groups? That is, ...
0
votes
1answer
142 views

Concerning functions (maps) and n-fold composition

I'm hoping to understand what the theorem below is saying and maybe receive hints but not the full proof. Apparently group theory can be used and I see that the function $f$ is closed as groups ...
1
vote
1answer
74 views

Order of the factor group by the center equals order of the group

My question is: What are the implications if the order of the factor group $G/Z(G)$ is equal to the order of the group $G$? I know that $|G/Z(G)|=|G|$ if $|Z(G)|=1$ meaning the center is trivial. ...
0
votes
1answer
53 views

How to show that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$?

I am reading the lecture notes. On Page 16, Line 1, it is said that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$. What does "factor through" mean? How to show that the ...
4
votes
1answer
72 views

Why $G/N$ is discrete?

I am reading the lecture notes. On page 15, Line -5, why $G/N$ is discrete? Thank you very much.
3
votes
1answer
245 views

Prove that a doubly transitive group is primitive.

My Attempt: (a) $S_n$ is transitive on $\{1,2,\dots,n\}$ and for any $(i,j)\in G_a,~(i,j)i=j$ whence $G_a$ is transitive on $\{1,2,\dots,n\}-\{a\}.$ (b) Without loss of generality let $|A|\ge2.$ ...
2
votes
1answer
63 views

How many subgroups of $D_4$ are isomophic to $V_4$

$D_4$= Dihedral Group, $V_4$= Klein Four Group. All subgroups of $D_4$: {$1$}, {$1,a$}, {$1,ab$}, {$1,ab^2$}, {$1,ab^3$}, {$1,b^2$}, {$1,b,b^2,b^4$}, {$1,b^2,a,ab^2$}, {$1,b^2,ab,ab^3$}, ...
2
votes
1answer
253 views

Transvection matrices generate $SL_n(\mathbb{R})$

I need to prove that the transvection matrices generate the special linear group. I want to proceed using induction on n. I was able to prove the 2x2 case, but I am having difficulty with the n+1 ...
1
vote
1answer
75 views

How to show that two representations are equivalent?

I am reading the lecture notes. On page 14, example of $C_{c}^{\infty}(G)$. I am trying to show that the map $A$ takes $f$ to $g\mapsto f(g^{-1})$ is an invertible element of ...
0
votes
1answer
119 views

Free abelian subgroup of index 2.

Let $G$ be a group with the following presentation $G=gp(x,y \mid x^2=y^2=1)$. I need to know, what further information about $G$ can be derived from knowing that $G$ has a free abelian subgroup of ...
1
vote
1answer
75 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
1
vote
1answer
59 views

How to show that $\pi^*(g)=\chi(\det g)^{-1}$?

I am reading the lecture notes. On page 14, how could we show that $\pi^*(g)=\chi(\det g)^{-1}$? I think that $\langle \pi^*(g)\lambda, v \rangle = \langle \lambda, \chi(\det g)^{-1} v \rangle = ...
1
vote
1answer
203 views

sufficient and necessary condition for Normality of a subgroup

Question is to : Prove that a subgroup $N$ of a group $G$ is Normal iff $gNg^{-1}\subseteq N$ for all $g\in G$. But, we define $N\unlhd G$ if $gN=Ng$ i.e., $gNg^{-1}=N$. So, question should be some ...
12
votes
2answers
975 views

Subgroups of a direct product

Until recently, I believed that a subgroup of a direct product was the direct product of subgroups. Obviously, there exists a trivial counterexample to this statement. I have a question regarding ...
2
votes
3answers
106 views

nonisomorphic groups whose quotients are isomorphic

Assume that we have two groups $A$ and $B$ such that $C \subset A$ and $C \subset B$ where $C$ is a normal subgroup of both $A$ and $B$. If we have that $A/C \cong B/C$ is it true that $A \cong B$? I ...
2
votes
1answer
78 views

filling in the gaps of this proof involving the fundamental theorem of finitely generated abelian groups [duplicate]

I have an abstract algebra proof I can not complete. The proposition goes as follows: Consider a group $G$ of order $m$. If $n$ divides $m$, prove that $G$ has a subgroup of order $n$. This question ...
1
vote
1answer
83 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
2
votes
2answers
455 views

Burnside's formula on a hexagon

I was trying to use Burnside's formula on a regular hexagon. I believe the answer is $13$, but I am trying to show my work. Here is what I have figured out. $$\frac{1}{12}(2^6 + 36 + 30 + \cdots?)$$ ...
1
vote
1answer
55 views

$G=C_{p}\times C_{p^2}$, describe $\mathrm{End}(G)$

I was asked to describe the group of the endomorphism of $G=C_{p} \times C_{p^2}$, with p prime ($C_n$ is the cyclic group of order $n$). I started setting (g,1) and (1,h) as generators of the ...
0
votes
1answer
231 views

Dihedral group as a direct product

In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two ...
2
votes
4answers
569 views

Need help understanding product of cycles

I need to understand how product of cycles work. The textbook i am referring gives explanation only for simple products and just answer for bigger ones. i would be very thankful if someone can explain ...
0
votes
4answers
181 views

$S_4$ does not have a normal subgroup of order $8$

Question is to prove that : $S_4$ does not have a normal subgroup of order $8$ I do not have any specific idea how to proceed but: Assuming there exists a normal subgroup $H$ of order $8$ in ...
0
votes
1answer
53 views

Questions about reductive groups.

I am reading the lecture notes. Let $G$ be a reductive group and $(\pi, V)$ a representation of $G$. For $v \in V$, define $\operatorname{Stab}(v)=\{g\in G \mid \pi(g)v=v\}$ and $V^{\infty}=\{v\in V ...
1
vote
1answer
3k views

Normal subgroups of dihedral groups

In relation to my previous question, I am curious about what exactly are the normal subgroups of a dihedral group $D_n$ of order $2n$. It is easy to see that cyclic subgroups of $D_n$ is normal. But ...
2
votes
0answers
72 views

(some) Dihedral groups have a right transversal isomorphic to some dihedral group.

Let $D_{2n}=\langle a,b;a^n,b^2,(ba)^2\rangle$ denote the Dihedral Group of order $2n$. $H=\langle a^m,b\rangle$ be a subgroup of $D_{2n}$ of even index $m$ such that $m$ divides $n$. Choose a right ...
0
votes
2answers
247 views

All pairwise non-isomorphic abelian groups of order 67500?

$67500=2^2*3^3*5^4$ => $2*3*4=24$ pairwise non-isomorphic abelian groups. Is this correct?
0
votes
1answer
61 views

Direct product of locally supersoluble groups

I'd like to know if the direct product of two locally supersoluble groups is locally supersoluble. Thank you in advance for the answer.
3
votes
2answers
143 views

Groups with $\wedge$-irreducible trivial subgroup

Suppose $G$ is a group satisfying the following condition: $$H \cap K = \{1\} \implies H = \{1\} \;\text{ or }\; K=\{1\}$$ for any two subgroups $H$, $K$, i.e. the trivial subgroup is ...
0
votes
1answer
441 views

left regular representation(group action) of a finite group $G$ Dummit Foote 4.2.11

Let $G$ be a finite group and let $\pi : G\rightarrow S_G$ be the left regular representation. Question is to : Prove that if $x$ is an element of order $n$ and $|G|=mn$ then $\pi(x)$ is a product ...
2
votes
1answer
111 views

Zero divisor conjecture

Let $K$ be a field and $G$ be a group. Then $K[G]$ is a domain iff $G$ is torsion-free. I know that "$\Leftarrow$" is conjectured to be always true. But what about the other direction?
4
votes
1answer
91 views

abstract algebra - a proof concerning a torsion group

Consider a group $G$. Suppose every element of $G$ has finite order. I want to prove that a finitely generated abelian group of this kind is finite. This idea of every element of $G$ having finite ...
3
votes
1answer
216 views

Using the fundamental theorem of finitely generated abelian groups

Let $H$ be an abelian group of order $k$. If $l$ divides $k$, prove that $H$ has a subgroup of order $l$. Noting that I am wanting to use the fundamental theorem of finitely generated abelian groups. ...
2
votes
1answer
200 views

Geometric interpretation of the dihedral subgroups of a dihedral group

I learned that a subgroup of $D_n = \langle r,s \mid r^n=s^2=(rs)^2=1 \rangle$, the dihedral group of order $2n$, is either cyclic or dihedral itself, and that a subgroup of the latter kind is of the ...
2
votes
2answers
408 views

What is the size of the normalizer of a subgroup generated by a $p$-cycle in a symmetric group?

Question: What is the size of the normalizer of a $p$-cycle (prime $p$) in the symmetric group $S_n$ ($n \geq p$)? If $n<2p$, we can actually find the size $N:=|N_{S_n}(\langle (12\cdots p) ...
2
votes
1answer
71 views

Matrix generating $\operatorname{SL}_n(\mathbb{R})$

How do I show that the following matrices generate $\operatorname{SL}_2(\mathbb{R})$ $\begin{pmatrix} 1 & a \\ 0 & 1 \\ \end{pmatrix}$ or $\begin{pmatrix} 1 & 0 \\ a & 1 \\ ...
2
votes
2answers
192 views

Isomorphism of algebras $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$

I have these two algebras and I need to know if they are isomorphic: $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$ Are there some general tricks how to deal with this type of tasks?
2
votes
1answer
452 views

Show that a group can not be expressed as union of two of its proper subgroups [duplicate]

Show that a group can not be expressed as union of two of its proper subgroups. I am not sure how to start.
2
votes
2answers
1k views

In an abelian group, the elements of finite order form a subgroup.

I need to show that elements of finite order in an abelian group form a subgroup of that group. Where do i start ?