Recent content by hgj

I'm really stuck on these two questions, please help! 1. Let G={invertible upper-triangular 2x2 matrices} H={invertbile diagonal matrices} K={upper-triangular matrices with diagonal entries 1} We are supposed to determine if G is isomorphic to the product of H and K. I have concluded...

I need to prove that the center of the product of two groups is the product of their centers. If I let G and H be two groups, then from definitions, the center of G is Z(G)={z in G such that zg=gz for g in G} and the center of H is Z(H)={z in H sucht that zh=hz for all h in H}. Also, the...

The question is to determine the group of automorphisms of S3 (the symmetric group of 3! elements). I know Aut(S3)=Inn(S3) where Inn(S3) is the inner group of the automorphism group. For a group G, Inn(G) is a conjugation group (I don't fully understand the definition from class and the...

I'm having trouble understanding how to represent a matrix on the (x,y)-plane. It seems like my classes expect me to know this, but I was never shown how. I've heard that there are a lot of connections between matrices and geometry, that matrices can be used to provide a geometric representation...

Okay, here's what I have now: Let T be a solution of AX=B. Then W+T is the set of solutions of AX=B. So AT=B. Then AX=AT <=> A(X-T)=0 <=> X-T \in W <=> X \in W+T I think I don't fully understand the definition of coset, so I'm not sure what to do from here. Our definition is : A left...

I need to do the following question: Let W be the additive subgroup of R^m of solutions of a system of homogeneous equations AX=0. Show that the solutions of an inhomogeneous system AX=B forms a coset of W. I really just don't know where to start. Any help would be appreciated.

if (ab)^l^c^m^(^m^,^n^) = e, doesn't that mean that the order of ab is the lcm(m,n)? Then, because a number always divides itself, that would explain why the order of ab divides the lcm(m,n). Is that right?

Okay, with the second problem about the order of the product of ab, I think it's that the order of ab divides the lcm(m,n) where m and n are the orders of a and b. I'm not sure how to prove this though.

We have not done Lagrange's Theorem yet, though I've looked at it and wish we had...it seems it would be helpful. I'm still not sure what to do for the second problem. I don't see what multiplying (ab) my multiples of m would do for me.

1. Describe all groups G which contain no proper subgroup. This is my answer so far: Let G be a such a group with order n. Then the following describe G: (a) Claim that every element in G must also have order n. Proof of this: If this wasn't true, the elements of lower order (elements of...

I see if i let x_1x_k = x_2x_k = \cdots = x_nx_k then, through the cancellation law, I get x_1 = x_2 = \cdots = x_n. Then, because it is given that S has an identity element, if these are all the elements of S and they are all equal, then they must all equal that identity element. So, each...

Here's the question: Prove that a nonempty subset H of a group G is a subgroup if for all x and y in H, the element xy^(-1) is also in H. We have a theorem that says if G is a group and H is a nonempty subset of G, then H is a subgroup of G iff: (1) H is closed (2) if h is in H, then...

Group Theory, please help! Okay, so I'm stuck on a couple questions from my homework, and any guidance would be much appreciated. 1. Prove that if G is a finite group with an even number of elements, then there is an element x in G such that x is not the identity and x^2 = e. I know...

If A is an nxn matrix, then detA = a11det(A11) - a21*det(A21) + ... + (-1)^(n+1)*an1*det(An1) (sorry, I don't know how to make things subscripts on this, so the 11, 21,...,n1 are supposed to be subscripts) That's the definition we're using for a determinant.

Okay, I need to prove that det(A^t) = det(A). I can see that it's true because I know columns and rows are interchangable (meaning you can use columns or rows when taking determinants), but I don't know how to prove this fact. Any help would be very appreciated.