Four Modern Algebra Problems

Date: 08/14/97 at 09:48:19
From: Cat Fyle
Subject: Modern Algebra problems

I have my final next week in Modern Algebra.  I have been lost in that
class since day one but have managed to maintain a B with your help.  
Could you please get me started on the following homework problems: 

1) a) Show that (1-i)^2i = (2^i)(e^1.570796)
   b) For all points z = x+iy in the right half-plane, (i.e.,x>0); 
      show that the principal value of ln(z) equals:

      ln(z) = 1/2ln(x^2 + y^2) + itan^-1(y/x).

2) Prove the following statement: If the integers a and b are 
   relatively prime, then there exist integers m and n such that 
   1 = ma+nb.

3) Let H be a subgroup of a finite group G. Show that the order of H 
   is a divisor of the order of G.

4) Let G be a set of four elements that is closed under an associative
   binary operator which satisfies the conditions:

   a) There exist and element e in G such that a*e = a for all a in G.
   b) Given that a is in G, there exists a mapping f(a) = b in G such 
      that f(a)*a = b*a = a.  Is this set a group?

Thank you for any help you can give me, CAT.

Date: 08/15/97 at 13:47:40
From: Doctor Rob
Subject: Re: Modern Algebra

>1)  a) Show that (1-i)^2i = (2^i)(e^1.570796)

Write 1-i in the form r*e^(i*t) for some r and t.  Do the same for 2.
Now substitute the first one in the left side and the second one in 
the right side of the above equation.

>    b) For all points z=x+iy in the right half-plane, (i.e.,x>0); 
>show that the principal value of ln(z) equals:
>ln(z) = 1/2ln(x^2 + y^2) + itan^-1(y/x).

Write z = x + i*y in the form r*[cos(t) + i*sin(t)] = r*e^(i*t) for 
some r and t. Now take ln of the resulting expression, and back-
substitute to get rid of r and t, replacing them with expressions 
in x and y.

>2) Prove the following statement:  If the integers a and b are 
>relatively prime, then there exist integers m and n such that 
>1 = ma+nb.

Let d be the smallest positive integer of the form m*a + n*b (this
exists by the Well-Ordering Principle: every nonempty set of positive
integers has a least element; you have to show that the set of 
positive integers of this form is nonempty). Divide a = q*d + r, 
0 <= r < d.

Then r = a - q*d = a - q*(m*a + n*b) = (a - q*m)*a + (-q*n)*b is of 
the same form. Since 0 <= r < d, and d was the smallest positive one, 
we must have r = 0, so a = q*d.  Similarly b = s*d.  Then d is a 
common divisor of a and b. Since a and b are relatively prime, d = 1. 

>3) Let H be a subgroup of a finite group G.  Show that the order of 
>H is a divisor of the order of G.

Consider the left cosets of H:  1H, g[2]H, ..., g[k]H.  Show that they
are all the same size, since they can be put into one-to-one 
correspondence with H.  (You have to say what the correspondence is, 
and prove that it is one-to-one and onto.)  They are disjoint (prove 
this), and together their union is all of G.  There are k of them, all 
of size #H, so k*(#H) = #G, and #H | #G.

>4) Let G be a set of four elements that is closed under an 
>associative binary operator which satisfies the conditions:
>a) There exist and element e in G such that a*e=a for all a in G.
>b) Given that a is in G, there exists a mapping f(a)=b in G such that
>   f(a)*a=b*a=a.  Is this set a group?

Are you sure that the last equation shouldn't say "f(a)*a=b*a=e"?  I
will call this (b').

You need to show that the group axioms are satisfied. You already have 
closure and associativity. You need identity and inverses. The natural
candidate for the identity is e and for the inverse of a would be 
f(a).

You are given by (a) above that e is a right identity. You need to 
show that e is also a left identity.

You are given by (b') above that f(a) is a left inverse of a. You need 
to show that f(a) is also a right inverse of a.

Alternatively, you could construct an operation table for * which
demonstrates that G doesn't have to be a group.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/