Abstract Algebra Answers

Assume that (G,*) is a cyclic group and that H is a subgroup of G. Explain why H is a normal subgroup G,and prove that G/H is cyclic.

Find all Abelian groups of cardinality 36, and all Abelian groups of cardinality 54.

Show that( R [x] /<x²+1>) isomorphic To C as fields.

Which of the following statements are true and
which are false ? Give reasons for your answers.
(a) Characteristic of a finite field is zero.
(b) Z12 is a field.
(c) In a ring with unity the sum of any two units
is a unit.
(d) Every element of Sn has order at most n.
(e) There is no non-trivial group homomorphism
from a group of order 5 to a group of order 6.

Let R = Z + √2 Z and S = the ring of 2 x 2
matrices of the form [
a 2b
b. a
. ] Where a,b belongs to Z

Show that f : R --> S defined by
f (a + √2 b) = [
a. 2b
b. a
] is an isomorphism of rings.

Let S be the set of all real numbers except
—1. Define an operation + on S by
x+y=x+y+xy; x,y belongs to S. Show that
< S, +> is an abelian group. Find a solution
of the equation 1+x = 2 in S.

Define a relation R on the set of integers Z
by R= {(n, n+ 3k) I k belongs to Z}. Show that R is an equivalence relation. Also find all distinct
equivalence classes.

Assume that (G,*) is a group, and that H={a is an element of G: a=a^-1 }. H is a subgroup of G. Prove that H is a normal subgroup of G