Linear Algebra and Group Theory

I am learning abstract algebra. Sorry to put so many questions here. Look forward to your help.Thanks a million 1.Prove any group of order 765 is abelian Prove that a group of order 135 has a normal subgroup of order 15 2.Let A be an abelian group with no elements of infinite order. Suppose that every element of prime order is of order 3. Show that the order of every element is a power of 3. 3.If G is finite and G/(A^B) isomorphic to (G/A) X (G/B) , prove that AB= G (A^B means the intersection of A and B) 4.If S is a simple nonabelian group, prove that contains a subgroup isomorphic to S 5.Give an example of fields such that K a normal extension…

Let k[x^2,x^3] be the subring of k[x] generated by the field k and the elements x^2 and x^3 then how to see that every ideal of k[x^2,x^3] can be generated by two elements? (It's said that one of the generator can be chosen as the polynomial with smallest degree) thx a lot~ Rp~

Can anyone tell me the difference between maximal and maximum subgroups of a semigroup, if there is one at all? I seem to be a bit confused about if there is a difference at all. In my lecture notes I can only find references to maximal subgroups, whereas on a problem sheet I can only find references to maximum subgroups. I also vaguely remeber my lecturer emphasining a difference between maximum and maximal, but I was concentrating on the proof she'd just shown at the time, so I'm not sure if she was talking about a difference within semigroup theory or the difference between semigroup theroy and group theroy. Any clarification would be great

I'm having trouble proving the following: Let [math]G[/math] be a group and let [math]a\in G[/math]. Then [math]\lambda_{a} \colon G \to G[/math] [math]g \mapsto ag[/math] defines a bijection For a group [math]G[/math], the family of bijections [math]\{\lambda_{g} \colon g \in G\}[/math]. The map [math]\lambda \colon G \to SymG[/math] [math]g \mapsto \lambda_{g}[/math] is an injective group homomorphism i can do the lambda being bijective part. but i can't show the map from G to Sym G defined as above is an injective homomorphism.

Here are two statments, the second one with a different order of the quantifiers. i understand what the first one says. can someone say what the second one means. First [math]\forall_{\varepsilon>0}\exists_{N\in\mathbb{N}}\forall_{n\ge N} |x-x_{n}|<\varepsilon[/math] Second. [math]\exists_{N\in\mathbb{N}}\forall_{\varepsilon>0}\forall_{n\ge N} |x-x_{n}|<\varepsilon[/math]

If R is a ring with unit element 1 and f is a homomorphism of R into an integral domain R' such that I(f) /= R. Prove that f(1) is the unit element of R'. for 1, a in R, 1.a = a.1 = a. Then f(1.a) = f(1)f(a) = f(a)f(1)= f(a) ( since R' is commutative) so we see that f(1) is the unit element of R'. Please check my proof and let me know of any mistakes I made. thanks in advance.

will someone please explain to me, for example in S_3, how do we find all the possible permutations, then from that how we find those permutations ( even ones) that are in A_3. I am looking at an example and it says that A_3 = {(1), (1,2,3), (1,3,2)}, but how can be (1,2,3), (1), and (1,3,2) be even permutations? Isn't the order of (1,2,3) = 3 ? I wrote down all the possible permutations of S_3, which are 6. Then I wrote them as cycles, so I got: (1)(2)(3) , (1 2)(3), (1 3)(2), (2 3)(1), (1 2 3), (1 3 2). Can someone tell me exactly how I can find the order of a permutation, and how I can find the elements of A_3. I know that A_3 will have 3 elements, since its order is…

Hello everyone, I am trying to find the form of the binomial theorem in general ring; that is, find an expression for ( a+b)^n, where n is a positive integer. I tried to solve it by induction, but usually we use induction when we know the answer and we try to show it is true. But here, I don't think here this will work. So I tried to write few terms, for n =1 a +b for n = 2 we have a^2 + 2ab + b^2 = a^2 + ab + ba + b^2, and for n = 3 it is a^3 + aab + aba + abb + baa+ bab + bba + b^3, so it is like the terms of (a+b)^n is consisted of all the possible arrangements between a and b in a sum, and every term of the sum has n entries either all a, all b, or a combination of…

Suppose n is an integer and n>=2. Show that An is a normal subgroup of Sn and compute Sn/An. That is, find a known group to which Sn/An is isomorphic.

Hello all..I am taking abstract algebra class and I am practicing problems dealing with ring theory. One of the problems I have is not being able to construct examples or provide counter examples. For some reasons, most of the counter examples turn out to be either S_3 or group of quaternions. Are there famous groups that one needs to know in order to be able to construct counter examples to famous theorems in group and ring theory? Also, I wonder if someone knows an example of an integral domain which has an infinite number of elements, yet is of finite characteristic?

Find all the homomorphisms from S4 to Z2? Find all the homomorphisms from S4 to Z3? Where do I start on this one? I know that one is that everything can be mapped to the identity in Z2 or Z3, but there where do I go? Help!

Let [math]F\colon \mathcal{A}\to \mathcal{A}'[/math] be an affine linear map, and let [math]\mathcal{C}[/math] be a convex subset of [math]\mathcal{A}[/math]. Show that its image [math]\mathcal{C}'=F(\mathcal{C})[/math] is a convex subset of [math]\mathcal{A}'[/math] Any Hints on how to proceed?

for sigma = 1 2 3 4 5 6 7 8 9 10 3 4 10 5 7 8 2 6 9 1 i know how to find the cycles, transpositions, & inverse but not the order of sigma, will somebody pls show/explain how, thanks

I need to prove that if T is regular matix than: ((T^-1)AT)^n=(T^-1)(A^n)T

Question: Prove that every group of order p^2 (where p is a prime) is abelian. What I started to say was this: |G| = p^2, then |G/Z(G)| = |G|/|Z(G)| which can equal p^2, p or 1, since those are the divisors of p. If |G/Z(G)| = p, then G/Z(G) is cycic because one element has order 1 and the rest of order p, which means there is a generator, it is cyclic and then G is abelian. If |G/Z(G)| = 1, then G=Z(G) so it much be abelian. However, the case where |G/Z(G)|=p^2 is where I get a little hung up. I know it never happens, but how can i prove it?

Question: Show that S4 has a unique subgroup of order 12. So far I know that A4 is a subgroup of order 12. So I was trying to prove by contradiction that there was another subgroup of order 12, say G. So the (|G||A4|/the intersection of G and A4) = 24. Then I need to show something to the effect that the intersection of G and A4 has to be a subgroup of order 6 in A4 and that that is a contradiction. This is as far as I got and my teacher says I am on the right track, but I have no ideas of where to go next, help please!

I've decided to write my essay (see the other thread that's hanging around somewhere or other) on the gamma function and Euler's constant. However, I am have much difficulty in finding applications of these concepts anywhere else in mathematics other than pretty abstract stuff. If anyone knows any good books/articles/journals that I can have a look at, it would be much appreciated. Cheers.

Can someone tell me what the origin of zerois???

let G be an agelian group such that the operation on G is denoted additively. Show that {a is an element of G| 2a = 0} os a subgroup of G. Compute the subgroup for G =13. let G be an abelian group. show that the set of all elements of G of finite order forms a subgroup of G. in addition to helping with these questions can you pls explain an abelian group, i know it is commutable but what does that actually mean. also what does additively mean? is the main idea to show 1) closed 2) identity 3) inverse to prove it is a subgroup? thanks so much!!

can somebody pls help with the following: Let G={1, -1,i,-i} where i^2 = -1. Show that G is an abelian group and give the group table.

I actually spend about 2hrs on this question.Would sombody please help me out. Ques: Let G be a group and H a subGroup og G.Let a,b be elements of G. prove : If a is an element of Hb, then Ha = Hb I Know that I have to show that Ha is a subset of Hb and vice-versa but I am unable to get there

I need to show that the only ideals of a field are the trivial one {0} and f itself. I'm not sure where to begin, or if there are any little tricks to use. So far I've thought about picking an ideal and assume it has an element not equal to 0. (If it doesn't have such an element then it is {0} an we are OK.) With this non-zero element of the ideal I would then add it to a non-zero element of the field. This too is in the ideal. (Am I right in assuming that any ideal of ring an hence a field is a subring under addition?) Since a field has inverses I could pick such an element out of it above so r+f=1, where r is in the ideal and f in the field. Then we …

Let H be a normal subgroup of G and let a belong to G. If the element aH has order 3 in the group G/H and H has order 10, what are the possibilities for the order of a?

I need to prove that if a,b, a is not 0, in Zp than there is n that b = na how should I start proving that ?

Let G =Z4 external direct product Z4, H= {(0,0),(2,0),(0,2),(2,2)} and K=<(1,2)>. Is G/H isomorphic to Z4 or Z2 external direct product Z2? Is G/K isomorphic to either?