0
votes
1answer
22 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
1
vote
0answers
12 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
3
votes
3answers
24 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
2
votes
1answer
30 views

Finding Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$. The mistake in this method?

We need to find the Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$ . This method does not give me the right answer (i.e $6$ ) . Attempt: We need to find the number of Cyclic ...
3
votes
1answer
51 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$ where $|a|=m_0$ and $|b| = m$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some ...
5
votes
4answers
90 views

Do we have $(G/H)\times H \cong G$ for groups in general?

After some thought I began to suspect $(G/H)\times H \cong G$, so I tried to construct an isomorphism by hand. I came up with $\varphi: (gH, h) \mapsto gh$ which came out to work provided $G$ is ...
0
votes
1answer
19 views

the smallest quasigroup, which is not a group

I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?
2
votes
1answer
40 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
1
vote
1answer
54 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
0
votes
1answer
24 views

Range and kernel of groups

Let $f: G \rightarrow H$ be a homomorphism. If the range of $f$ has $n$ elements, then $x^n \in$ ker $f$ for every $x \in G$. I can kind of understand why this is true. The ker of $f$ is $\{x \in ...
1
vote
1answer
37 views

Cyclic and abelian group

A group $G$ has order $25\cdot 47\cdot 17$. Is it cyclic and/or abelian? I know that a group of order $47$ or $17$ is cyclic, should I somehow use it?
0
votes
1answer
21 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
2
votes
0answers
29 views

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. Condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle ?$

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. State a necessary and sufficient condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle$ Attempt : Let $l = ...
2
votes
1answer
62 views

Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
3
votes
2answers
62 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
6
votes
4answers
216 views

How is this subgroup normal?

Let $G$ be a group, and let $U$ be a subset of $G$. Let $\hat{U}$ be the smallest subgroup of $G$ containing $U$. Then $\hat{U}$ is the intersection of the collection of all the subgroups of $G$ ...
2
votes
1answer
31 views

Surjective Homomorphism Symmetric group

For $G=S_4$ i'm having a bit of trouble following the solution. For the blue underline I was wondering if there is a strategy for spotting this relatively quickly. For the green underline I ...
3
votes
0answers
38 views

If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...
0
votes
0answers
18 views

determining maximum number of elements of particular order

In direct product of an infinite group, say of nonzero reals or positive reals, is there a way to determine a number of elements of a particular order (e.g., 1 or 2), or at least know whether the ...
2
votes
1answer
28 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
0
votes
1answer
22 views

order of an element formula

I was wondering whether there's a formula or something. If it is given that $x^n = e$ and $x^m = e$, does it mean $x^{gcd(n,m)} = e$, so we can determine whether $x=e$ or $x \ne e$?
1
vote
4answers
93 views

proof a function is an isomorphism

When we prove a function is an isomorphism, we need to prove it's a bijection and it's closed under an operation. In one example I had no problem proving the first part, but in the second part, I ...
1
vote
1answer
35 views

How do I show a mapping is a homomorphism?

I don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and more ...
0
votes
1answer
18 views

order of elements in direct product

I have a conceptual question: if a group has 1 element of order 1 and 1 element of order 2 (e.g., nonzero reals), what changes if your take its direct/cartesian product as a group?
9
votes
2answers
346 views

Is the Axiom of Choice implicitly used when defining a binary operation on a quotient object?

Let's say you have a group $(G,\cdot)$ and you have a normal subgroup $N$ (note we are considering this only as a set). And now we want to define a binary operation $\star$ on $G/N$ such that $(G/N, ...
0
votes
2answers
40 views

Subgroup of size p p=prime

Are subgroups of size $p$ where $p$ is prime, cyclic subgroups? I understand that if the group $G$ is prime order, then $G$ is cyclic.
-1
votes
1answer
29 views

Surjective Homomorphism Dihedral Groups [on hold]

For the first underlining i'm just wondering where i have used the fact that $\phi$ is surjective. For the second underling I don't understand the explanation it gives for why $<p^2>$ is the ...
5
votes
2answers
78 views

Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups?

Well, this is my question. Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups (maybe one being trivial)? Thanks!
1
vote
3answers
63 views

Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am lookking for ...
1
vote
1answer
35 views

Necessary and sufficient for $\operatorname{orb}(x)=\operatorname{orb}(y) \iff \operatorname{Stab}(x)=g\operatorname{Stab}(y)g^{-1}$

Are orbits equal if and only if stabilizers are conjugate? You may get some insights from the link above. My Question: What is the necessary and sufficient condition for the above statement to be ...
1
vote
0answers
16 views

Meaning behind the conjugacy class in describing geometry of solids

When we consider the group of rotataional symmetry , say of a cube or a dodecahedron, it is not difficult to see the symmetry group is isomorphic to a $S_4$ , $A_5$ respectively. Mooreover, when you ...
2
votes
0answers
26 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
2
votes
3answers
134 views

Can we find an element of infinite order in a symmetric group of infinite order?

In particular, I'm thinking of a simple example: the group $S_\Omega$ given $\Omega = \{1, 2, 3, ...\}$. I've been thinking of elements of $S_\Omega$ in terms of their cycle decomposition, which may ...
1
vote
0answers
42 views

Is there a simple proof for Fundamental theorem of finitely generated abelian group?

I'm studying two abstract algebra texts simultaneously now, namely, 'Dummit&Foote' and 'Fraleigh'. Both of these texts introduce 'Fundamental Theorem of finitely generated abelian group' without ...
1
vote
2answers
36 views

Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
2answers
26 views

one to one mapping from $A(s_1)$ into $A(s_2)$

Let $S_1$ and $S_2$ be two sets. Suppose there exists a one to one mapping $\phi$ of $S_1$ into $S_2$. Show that there exists an one to one mapping from $A(S_1)$ into $A(S_2)$, where $A(S)$ means the ...
0
votes
0answers
32 views

Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
1
vote
1answer
31 views

Group under composition $\circ$?

Define the set of all affine real-valued functions $G:=\{f_{a,b} \mid a,b \in \mathbb{R},a \neq 0\}$ where $f_{a,b} : \mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_{ab}: x \mapsto ax+b$. Is ...
0
votes
0answers
37 views

$\mathrm{SL}_2$ acts naturally on binary forms

Problem: Let $F^{(n)}$ be the vector space of binary forms of degree $n$ in two variables with coefficients in $\mathbb{C}$. Show that $\mathrm{SL}_2$ acts on $F^{(n)}$ and ...
2
votes
1answer
42 views

Representation of $GL_2$ on $K^2$

In one of my problems it says the following: Let $K$ be an infinite field. Consider the linear action of $GL_2$ on $K[x,y]$ induced by the natural representation of $GL_2$ on $K^2$. I don't know what ...
1
vote
1answer
41 views

Conjugacy Class Proof

I don't follow the answer to b). I understand the first couple of sentences, as they involve a lemma from the notes but i am lost thereafter. ...
1
vote
1answer
59 views

Sylow theorem application

Suppose that G is a group of order $60$ and $G$ has a normal subgroup $N$ of order $2$. Show that $G$ has normal subgroups of order $6$ and $10$ no of $3$ Sylow subgroups are 1 ,4 or 10 (by Sylow's ...
2
votes
1answer
17 views

Subgroup of order $p$ is normal

I am trying to show that an arbitrary group $G$ of order $p^n$ has a normal subgroup of order $p$. My first instinct is to say that by Cauchy's Theorem, there is some element $x \in G$ such that the ...
2
votes
1answer
35 views

Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& ...
0
votes
2answers
44 views

Conjugacy classes

I don't seem to be able to follow this part of the proof. Why are we able to say $x \sim y$? Why does it suffice to prove that $y \in H$? Let $G$ be group. $(a)$ We say that elements $x,y \in ...
5
votes
1answer
56 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
0
votes
1answer
49 views

Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
2
votes
1answer
60 views

When is $\langle x,y\rangle$ equal to $\langle x\rangle\langle xy\rangle$?

I would like to know necessary and sufficient conditions on $x$ and $y$ to have $\langle x,y\rangle=\langle x\rangle\langle xy\rangle$. Sure that: $\bullet$ $\langle x\rangle$ and $\langle ...
3
votes
2answers
50 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...