Abstract Algebra Answers

The set of cosets of (1 2)in S3 is a group with respect to multiplication of cosets.

Suppose T in L(V) and U is a subspace of V.

Prove that if U subset of null T, then U is invariant under T.

If H is a finite subgroup of a group G, for all

a

ϵ

G

aϵG the left coset aH and the right coset Ha coincide if

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.

Let

G

1

G1 and

G

2

G2 be two groups and

f

:

G

1



i

g

h

t

a

r

r

o

w

G

2

f:G1 ightarrowG2 be a homomorphism, then the kernel of f is define by

1. Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way.

2. i. State the Lagrange’s theorem of group theory.

ii. For a subgroup H of a group G, prove the Lagrange’s theorem.

iii. Discuss whether a group H with order 6 can be a subgroup of a group with order 13 or not. Clearly state the reasons.

Build a set of operation tables for group G with orders from 1, 2, 3 and 4 using the elements of a, b, c, and e as the identity element.

8.α+β=β+α

is said to be ____ under addition of vectors.

a.associative

b.distributive

c.commutative

d.Abelian

Under vector addition α+β+γ=γ+β+α

is called the ____ law.

a.associative

b.distributive

c.commutative

d.Abelian

If x∗y= x+y+xy, find 2∗1?

a.5

b.2

c.3

d.4