1. a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of (M(R),+)? If yes, prove it. If no, provide a counterexample. b) Consider the group (ZxZ, +). Is the subset K = {(x, y) | x+y> 0} a subgroup of (ZxZ,+)? If yes, prove it. If no, provide a counterexample. c) Consider the group (ZxZ, x). Is the subset D = {(x, y) | x + y = 0} a subgroup of (ZZZZ, X)? If yes, prove it. If no, provide a counterexample. NOTE: The operation here is multiplication. d) For the group (Z12, +), find all the cyclic subgroups. (ie. Find all where a Є Z12.)

(4) Prove that Z x Z/((2,2)) is an infinite group but is not an infinite cyclic group.

15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the image group.

4. (a) Let G be a group such that |æ| = 2 for every x # e. Prove that G is abelian. (b) Let G be an abelian group. Prove that the set of elements of G of finite order is a subgroup of G. (c) Consider the following elements of GL2(R): a = b = -1 Show that Ja| = 3, |6| = 4, but |ab| = ∞.