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 ...
3
votes
1answer
57 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
3
votes
5answers
217 views

Number of groups of a given order

In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$? I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but ...
1
vote
1answer
26 views

Showing if $H,K$ are subgroups of $G$, the $\varphi(hak)=a^{-1}hak$ is bijective.

Here's my problem...I have to ultimately show that $\varphi(hak)=\varphi(hbk)\Rightarrow hak=hbk$ and I'm having issues... I am told, both $H,K$ are subgroups of $G$ and $a\in G$. The goal is to show ...
0
votes
1answer
53 views

Modern Algebra: Groups

Is this the way to solve the question? Question a). Find the center of the group $S_3 \times \mathbb Z/6\mathbb Z$ Ans: $S_3$is the order of 6 element therefore, {1,(12),(23),(13),(123),(132)} and ...
1
vote
0answers
43 views

If $x^m=y^n=e$ and $(m,n)=1$, then $o(xy)=mn$ [duplicate]

If $G$ is a group with $xy=yx$ for $x,y\in G$, and $x^m=y^n=e$, then: (1) $o(xy)|mn$, where $o(xy)$ is the order of $xy$, and (2) if $(m,n)=1$ then $o(xy)=mn$. Well, I've already proved (1), but ...
0
votes
1answer
24 views

Induction with two indexes

I want to prove that if $G$ is a group and $a\in G$, $n,m\in \Bbb Z$, then $a^na^m=a^{n+m}$. I think, that it's easier to prove the case when $n,m\in \Bbb N$. I found this question: Induction (over 2 ...
2
votes
1answer
71 views

Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.

(Reference : Fraleigh, A first course in abstract algebra) Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, ...
1
vote
1answer
73 views

Group with an even number of elements.

If $G$ is a group such that $|G|=2n$. Prove that there's an odd number of elements of order 2, and then there's an element which is its own inverse, besides of the identity. If we consider all the ...
2
votes
2answers
74 views

Proving that this is not a group.

I got the set: $G=\{p/q\in \Bbb Q : (p,q)=1$, with $q$ odd number $\}$ and the binary operation $a*b:=a+b$. And I say that $(G.*)$ isn't a group because it doesn't have an identity. My proof is: We ...
3
votes
2answers
55 views

Proving uniqueness (basics of group theory)

If $(G,*)$ is a group, prove that the identity and the inverse elements are unique. What I did for the first one is: Suppose $\exists e,g\in G$ such that $\forall a\in G a*e=e*a=a$ and also that ...
1
vote
0answers
39 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
3
votes
2answers
54 views

Condition such that $\langle{S}\rangle=S$

So let $S\subseteq{G}$ where $G$ is a group and $S$ an arbitrary subset. Let $\langle{S}\rangle$ be the subgroup in $G$ generated by $S$. What is the condition such that $\langle{S}\rangle=S$? My ...
0
votes
3answers
64 views

Exercise: product of transposition

How would I go about computing $$(1 2 3)\cdot(12)(34)$$ I know the definitions but I do not know how to apply them here. This is rather strange and odd-looking to me. I know I have to construct a ...
2
votes
1answer
66 views

Suppose that the cyclic group $G$ acts on a set $S$

Suppose that the cyclic group $G$ acts on a set $S$ and $g_1$ and $g_2$ generate $G$. Show that $|$Fix $g_1|=|$Fix $g_2|$. We know that there is a function $\psi: G\times S\rightarrow S$ with the ...
3
votes
3answers
101 views

What math will I need in order to learn Microsoft's UProve?

I'm studying Microsoft's UProve (independent studies at 35 years old) and forget most of the Math I learned in college. I intend to proceed and learn the contents of this chapter of this book but can ...
3
votes
2answers
392 views

Factor groups and isomorphisms

I've somewhat recently been going back through one of my brother's old textbooks reviewing group theory. I'm up to a chapter called Factor-Group Computations and Simple Groups. The problems at the ...
16
votes
4answers
1k views

Number of subgroups of prime order

I've been doing some exercises from my introductory algebra text and came across a problem which I reduced to proving that: The number of distinct subgroups of prime order $p$ of a finite group ...
4
votes
2answers
463 views

Elementary Group Theory, Sylow Theorems

So it's time for me to post another elementary question. I've been stuck on this exercise for quite some time now, and really can't find a satisfactory solution for this. The section on applications ...
2
votes
0answers
222 views

Isomorphism of groups with certain property, p-groups

I'm doing exercises from Hungerford's book "Abstract Algebra: An Introduction". The exercise is in section 8.2, numbered 22. I would like someone to check my proof, as I have reasonable doubts that ...