Abstract Algebra Answers

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 f be a degree two polynomial over R in n-variables and a = (a1, . . . , an) a
critical point of f. Show that there is a homogenous degree two polynomial Q
such that
f(x) − f(a) = Q(x − a) (x ∈ Rn
)

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