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Flashcards in Alg Deck (68):
1

standard sets

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Z,Q,R,C,M_{mxn}(S), M_n(S)"+" superscript, "+" with overbar, *

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binary operation on S

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closed under an operation

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associative

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semigroup

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identity

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monoid

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inverse

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group

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defn, equivalently 4 conditions

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properties of groups

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6 of em

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commutative

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Abelian

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order of S

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examples of groups

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7 groups, 2 more

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general linear group

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special linear group

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nth roots of unity

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symmetric group on n letters

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dihedral group

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group table

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additive group of integers modulo n

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multiplicative group of integers modulo p

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cyclic group

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generator of (Z_n,+)

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when a^m is a generator of G

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subgroup

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center

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cyclic subgroup generated by a

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order of an element

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finitely generated group

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relations

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defn, Klein four-group presentation

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Lagrange's theorem

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Abelian subgroup of certain order

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subgroups of cyclic groups

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Cauchy's theorem

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Sylow's first theorem

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isomorphic

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direct product

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when Z_m+Z_n+... is cyclic

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elementary divisors of G

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invariant factors of G

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coset

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group homomorphism

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monomorphism

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epimorphism

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isomorphism

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endomorphism

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automorphism

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properties of homomorphisms

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6 of em

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kernel

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ring

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3 main conditions, total of 8 sub-conditions

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ring with unity

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commutative ring

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subring

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characteristic of the ring

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ring of integers

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ring of integers modulo n

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ring of Gaussian integers

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ring of real-valued functions on R

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ring homomorphism

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properties of ring homomorphism

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3 of em

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integral domain

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zero divisor

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zero divisors in Z_n

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integral domains Z_n

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unit

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division ring

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field

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defn, 6 conditions