Recent content by wnorman27

Well for b'\inB and c'\inC, b'1=b'(b+c)=b'b+b'c=b'b (since b'c is an element of B\capC, and is thus 0) so b'=b'b, and similar arguments show b'=bb' c'=c'c c'=cc' so that b is the unit in B and c is the unit in C. Given that we can write r=b+c for unique b and c, this and the rest...

My text defines a ring to be unital but not necessarily commutative, and an ideal to be a two-sided ideal unless otherwise stated.

Homework Statement If a ring R contains two ideals B and C with B+C=R and B\capC=0, prove that B and C are rings and R\congB x C. Homework Equations B+C={all b+c|b\inB and c\inC} The Attempt at a Solution So far I've discovered that if the unit of R is in one of the ideals then...

I understand that in the context of groups, this h' is just the result of conjugation of h by g, but my thought was that perhaps this might occur outside of the context groups (say in cases where inverses may not exist)? I can't think of any examples of this though... maybe this is just not useful.

Homework Statement I'm trying to figure out if the following property has a name: for g\in G, h\in H, \exists h'\in H s.t. gh=h'g. obviously this is not quite commutativity, but it seems like it might be useful in a variety of situations. Homework Equations I've just finished a proof that...

Let h\in\ H with h\neq e, and let k\in\ K with k\neq e. Then what can you say about khk^{-1}?

Hi Jbunniii, Thanks for all your help! I'm using Algebra (the advanced one) - as my primary text as I work through abstract algebra for the first time, but I've been reading parts of a few other algebra textbooks (Dummit+Foote, Fraleigh, Artin, and some course notes from a prof at uc...

|ST| = Sum of orders of distinct cosets of S of the form St for some element t. For two cosets St and St', we have that St=St' iff St*t'^{-1}=S, which means t*t'^{-1}\in S. But t*t'^{-1}\in T also, so we must have t*t'^{-1}\in (S\cap T). Then (S\cap T)t=(S\cap T)t' so that there are the same...

Homework Statement If S and T are subgroups of a finite group G, prove that [S:1][T:1] ≤ [S\capT:1][S\veeT:1]Homework Equations Notation: [A:B] is the number of cosets of B in A for some subgroup B of A. note that [A:1] is the order of A. Lagrange's Theorem: For some subgroup S of G...

define D(x,y,z) = (x-1,y-1,z-1) for x>0. define d(x,y,z) = (0,y-1,z-1) for x=0 notice that D^k(x,y,z)=(x-k,y-k,z-k) and d^k(0,y,z)=(0,y-k,z-k) now D^x(x,y,z) = (0,y-x,z-x) and d^(y-x)[D^x(x,y,z)] = (0,0,z-x-(y-x)) =(0,0,z-y) this suggests: a=-y+z b=-x+y c=x x=c y=b+c z=a+b+c the matrix |0...

Homework Statement Given some group G with generators g_{1},g_{2},...,g_{n} as well as a description of the action of g on the elements of some set S={s_{1},s_{2},...,s_{k}}, how in general does one go about finding a complete defining relations (and showing they are complete)? Homework...

So I'm trying to wrap my head around how I would find the surjective group homomorphism φ:Gn→Sn+1. Also I believe that statements about |Gn/H| must be using some results about the orders of groups that I haven't yet come across - I'll look into that. The bigger question I have though is how...

micromass, I can agree with your definition of complete, though I'm still lost as to how one would show that our group G = Sn from the given information. Also, t_{i} are given explicitly as t_{i} = (i i+1), so the fact that they are transpositions is clear. I see how Cayley's theorem...

Homework Statement I am working through MacLane/Birkhoff's Algebra, and in the section on Symmetric and Alternating groups, the last few exercises deal with generators and Defining relations for Sn (the symmetric group of degree n). These read: 11. Prove that Sn is generated by the cycles (1...