Date: 02/22/99 at 01:24:35 From: Mari Subject: Algebraic
Structures A friend and I are stuck on two homework problems.
Please help us. 1) Suppose that N and M are two normal subgroups
of G and that (N intersect M) = (e). Show that for any n element
of N, m element of M, nm = mn. We began by manipulating what we
wanted and have e = m^-1 n^-1 m n, where we have the 2 forms of
elements being in N and M. Is this a correct start? 2) Let G be
defined as all formal symbols (x^i y^j), i = 0, j =0, 2, 3, ...,
n-1, where we assume (x^i y^j) = (x^i prime y^j prime) iff i = i
prime, j = j prime x^2 = y^n = e, n>2 xy = y^-1 x. a) Find the
form of the product (x^i y^j)(x^k y^L) as (x^alpha y^beta). b)
Using this, prove G is a non-abelian group of order 2n. c) If n
is odd, prove the center of G is (e), while if n is even, the
center of G is larger than (e). For a) do we did cases applying
n > 2, n < 2 since we tried to substitute the given into what we
wanted, but we could not get it. For b) and c) we really did not
have a