Algebra assignment

I need the solutin for these problems

Name:
1. A group G is abelian if and only if the map o : G+ G given by (x) = 2-1 is an automorphism.
2. Let S be a nonempty subset of a group G and defiend a relation on G by a ~ b if and only if
ab-1 e S. Show that is an equivalence relation if and only if S is a subgroup of G.
3. If G is a group, then C = {a e G| ax = xa for all x € G} is an abelain subgroup of G. (C is
called the center of G).
4. Let f: G+ H be a homomorphism of groups, A is a subgroup of G, and B is a subgroup of H.
(a) Show that ker f and f-‘(B) are subgroups of G.
(b) Show that f(A) is a subgroup of H.
5. Let H and K be subgroups of a group G. Then HK is a subgroup if and only if HK = KH.
6. If p >q are primes, a group of order pq has at most one subgroup of order p (Hint: Suppose H,
K are distinct subgroups of order p. Show H NK =< e>.)
(b) If K is the cyclic subgroup (of order 3) of Ss generated by (2
(1 2
Then no left coset
21
of H (except H itself) is also a right coset. There exists a € Sz such that aH n Ha = {a}.
1 2 3
then every left coset
(2 3 1
of K is also a right coset of K.
8. Let G be abelian group of order pq, with (p, q) = 1. Assume there exist a, b e G such that
al = p, 161 = q. Show that G is cyclic.
9. Let Q: be the group (under ordinary matrix multiplication, generated by the complex matrices
A=
and B = where i2 = -1. Show that Q8 is nonabelain group of order 8
-1
(called the quaternion group). (Hint: Observe that BA = AⓇB, and every element of Qs is of
the form A’ Bi. Note also that A4 = B4 = 1, where I =
is the identity element of Qs.)
(62)
10. Let H be a group (under matrix multiplication) of real matrices generated by C =
(o
and D =
(1 o).
Show that H is a nonabelian group of order 8 which is not isomorphic to Qs,
but is isomorphic to the group D.