August 16th, 2012, 11:13 AM  #1 
Newbie Joined: Aug 2011 Posts: 3 Thanks: 0  Groups
Let (G,*) is group and A,B are her subgroups.  Prove that A?B is subgroup of (G,*).  If A=10 and B =7 how many elements A?B have and who are they?  Find subgroups A,B from (Z,+) so that A?B isn't subgroup of (Z,+). thanks 
August 23rd, 2012, 06:45 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Groups
Have you made no attempt at all yourself? It should not be difficult. Since A and B are 'subgroups' of G they are subsets of G and so is their intersection. So the only thing that remains to be shown is that the intersection is 'closed under the operation". If x and y are member so then they are both members of A and so x+ y is in A. Now do the same thing with B. "If A=10 and B =7 how many elements A?B have" is closed under the operation and so is a subgroup of B. The crucial point is that B= 7 is a prime number. What does that tell you about the subgroups of B? "Find subgroups A,B from (Z,+) so that A?B isn't subgroup of (Z,+)." I'll give you a hint. All subgroups of (Z,+) are of the form {in is is any integer, n is a fixed integer}. In other words, they are "all even numbers", "all multiples of three", etc. 

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