Abstract Algebra & Number Theory Summary

GRE Math Subject · Axiom Academy

SUMMARY Abstract Algebra & Number Theory Complete unit review: Groups, rings, fields, and number theory for the GRE Math Subject Test Abstract algebra and number theory together account for roughly 25% of the GRE Math Subject Test. This unit covers the algebraic structures (groups, rings, fields) and classical number theory results you need. The emphasis is on recognizing structure, applying key theorems quickly, and connecting algebraic ideas to concrete examples. A group is a set with an associative binary operation, an identity element, and inverses for every element. Key examples: under addition mod (cyclic group of order ) the symmetric group on elements ( ) invertible matrices under multiplication A subgroup is a subset closed under the group operation and inverses. A cyclic group is generated by a single element. Key facts: Every subgroup of a cyclic group is cyclic. The subgroups of are exactly for each divisor of . If is a subgroup of finite group , then divides . Corollary: The order of every element divides . A group of prime order is cyclic. A subgroup is normal ( ) if for all . The quotient group has cosets as elements with . GRE use: Every subgroup of an abelian group is normal. Kernels of homomorphisms are always normal. A homomorphism preserves the group operation. The First Isomorphism Theorem: Cayley's Theorem: Every group of order is isomorphic to a subgroup of . Sylow 1: A subgroup of order (Sylow -subgroup) exists.

This is the written version of the interactive lesson above. See the full GRE Math Subject course.