Abstract Algebra Answers

Let
A
=
l
e
f
t
−
1
,
1
,
−
i
,
i
r
i
g
h
t
,
i
=
s
q
r
t
−
1
.
A=left−1,1,−i,iright,i=sqrt−1. Let the binary operation be multiplication, what is the inverse element of
i
i?

find all finite abelian group of order less than or equal to 40 up to isomorphism.

Let A=left−1,1,−i,iright,i=sqrt−1. Let the binary operation be multiplication, what is the inverse element of i?
a.\\(-i\\)
b.\\(i\\)
c.1
d.\\(-1\\)

Subtraction is an operation on Z which is neither commutative nor___
a.Distributive
b.Commutative
c.Additive
d.Associative

Every element
x in mathbbZ has ____________ inverses with to respect to addition as a binary operation.
a.\\(-x\\)
b.\\(0\\)
c.\\(frac{1}{x}\\)
d.\\(x\\)

Check whether or not S = { a0 + a1x + .... + an(x^n) belongs to Z[x] | 5 | a0} is an ideal of Z[x].

Is { [a a+b a+b b] | a,b belongs to Z} a subring of M2(Z)? Why, or why not?

Check whether any group of order 44 has a
proper normal subgroup or not.

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.

show that the roots of the equation z^4-1=0 form a cyclic group of order 4.