Abstract algebra in the Encyclopedia.
solution Explain the "null set."
proof Prove that and , if and only if is a subgroup of .
solution Define actions, centralizers, normalizers, stabilizers, and centers.
proof Prove that if and then the following are equivalent:
(a) which means is a generator of .
(b) i.e. r and are relatively prime.
(c) such that .
solution Let and belong to the group . If and , where and are relatively prime, show that and that .
proof Let be any subgroup of the group . The set of left cosets of in form a partition of . Furthermore, for all if and only if and in particular, if and only if and are representatives of the same coset.
proof If G1 is a subgroup of G and H1 is a subgroup of H, prove is a subgroup of .
proof Let H and K, each of prime order p, be subgroups of a group G. If , prove .
proof If p and q are prime, show every proper subgroup of a group of order pq is cyclic.
proof Let G be a group such that (ab)2 = a2b2 for all . Prove G is abelian.
proof Let G be a group such that (ab)i = aibi for all and for three consecutive integers i. Prove G is abelian.
proof Prove that a group of order 312 has a normal Sylow p-subgroup.
Groups- facts and examples
- Addition of residue classes in is associative.
- Multiplication of residue classes in is associative.
- A finite group is abelian if and only if its group table is a symmetric matrix.
- If then .
- If then if and only if .
- If , then is abelian.
- is abelian if and only if both and are abelian.
- If and is not a power of , then .
- Every element of which is not a power of is of order 2.
- is a non-abelian group for all .
- Disjoint cycles commute.
- The order of a permutation is the l.c.m. of the lengths of the cycles in its cycle decomposition.
- An element has order 2 in if and only if its cycle decomposition is a product of commuting 2-cycles.
- Let be a prime. Show that an element has order in if and only if its cycle decomposition is a product of commuting p-cycles.
- If then the number of -cycles in is given by .
- is an matrix with entries from and nonzero determinant.
- is non-abelian.
Homomorphisms and Isomorphisms
- is a homomorphism if . This implies and .
- The exponential map defined by is an isomorphism from to .
solution In the ring , prove the following:
(a) if and only if
solution Show that is irreducible but not a prime in .
solution (Dis)prove: as rings, where is prime.
solution Show that an integral domain with a descending chain condition (if is a descending chain of ideals, then there exists such that ) is a field.
solution Let be a nilpotent element of the commutative ring . Let for minimal . Prove that is nilpotent for all .
solution Let be a nilpotent element of the commutative ring . Let for minimal . Deduce that the sum of nilpotent element and a unit is a unit.
solution Prove that the center of a ring is a subring containing the identity.
Chinese Remainder Theorem
Statement of theorem: Let be a ring with identity .
Let be ideals of . The map defined by is a ring homomorphism with kernel . If for each with the ideals and are comaximal, then this map is surjective and , so
solution Let be integers which are coprime to each other.
(a) Show that the Chinese Remainder Theorem implies that for any there is a solution to the simultaneous congruences
and that the solution x is unique mod .
(b) Let be the quotient of by . Prove that the solution in (a) is given by
(c) Solve the simultaneous systems of congruences