Abstract Algebra Answers

If T is nilpotent transformation on V of index m. Prove that m cannot exceed the dimension of V

Determine whether or not ∗ gives a group structure on the set. If it is not a group, say which

axioms fail to hold.

Define ∗ on Z by a ∗ b = ab.

let R be a ring in 1

1. let r be a ring with identity and let S be a subring of R containing the identity. prove that if u is a unit in S then u is a unit in R.

Cyclic Groups

1. Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.
2. List the elements of the <i>, i.e. cyclic subgroup generated by i of the group C* of nonzero complex numbers under multiplication.
3. Let G=<a> and |a|=24. List all generators for the subgroup of order 8.
4. Draw the subgroup lattice diagram for Z₃₆ and U(12).

Prove that (Q, +) is an abelian group under ordinary addition.

Under vector addition,

α+(β+γ)=(β+α++γ is called the ____ law.

Prove that SL(n,F) is a subgroup of GL(n,F)