Isomorphic Groups and SubringsDate: 04/15/98 at 15:15:07 From: Leonie Williams Subject: abstract or modern algebra I have a few problems in my abstract algebra class that I just can't figure out. The problem is that I don't even know how to get started on these. 1) Let G = {a+b*sqrt[2]: a,b element of Q (rationals)}, H = {Matrix((a 2b) (b a): a,b element Q} Show that G is isomorphic to H as additive groups. 2) Let M = {Matrix ((a -b) (b a))/ a,b element R (reals). Show that C (complex numbers) is isomorphic to M as additive groups. Also, delete 0 + 0i from C and Matrix ((0 0) (0 0)) from M and show that C* is isomorpic to M* under multiplication where * indicates deletion of "zero." 3) Show that {a + b*2^(2/3)+ c*4^(2/3): a,b element Q(rationals)} is a subring of R(reals). Date: 04/15/98 at 16:31:49 From: Doctor Rob Subject: Re: abstract or modern algebra To show two additive groups A and B are isomorphic, you have to construct a map f:A -> B, from one to the other. This map has to have several properties, which you would have to prove: (1) f is one-to-one: f(x) = f(y) implies x = y (2) f is onto: for every y in B, there is an x in A with y = f(x) (3) f is an additive homomorphism: for every x and x' in A, f(x+x') = f(x) + f(x') In the first problem, A is G, B is H, and the map is: f(a+b*sqrt[2]) = ((a 2b) (b a)) You have to show that the three properties are true for this function f. In the second problem, A is M and B is C, and the map is: f(((a -b)(b a))) = a + b*i Again you have to show that the three properties are true. To show that A = M* and B = C* are isomorphic as *multiplicative* groups, you need to replace (3) above with: (3') f is a multiplicative homomorphism: for every x and x' in A, f(x*x') = f(x)*f(x') For the third problem, you have to show that the set, call it S, is a subset of the real numbers (that's pretty easy!), and that the ring axioms hold in S. There is no problem with commutativity, associativity, or distributivity, because those properties are inherited from the real numbers. (If they are true for any real numbers, they must be true for these special ones.) The main things you have to show are that addition and multiplication of numbers in S give you results which are again in S (this is called closure under addition and multiplication), that the real number 0 is in S (the identity element for addition), and that the additive inverse of any element in S is also in S. To do this, you need to use the fact that r = 2^(2/3) and s = 4^(2/3) satisfy the equations r*r = s, r*s = 4, and s*s = 4*r. You also need to use the fact that Q is closed under addition, multiplication, and additive inverses. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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