Search results for “algebraic-geometry”

Accepted for publication in Algebra & Number Theory.

The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group $S_n$ — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.

The order of an element and of a group, the order of a permutation as the lcm of its cycle lengths; generators and cyclic groups; homomorphisms and isomorphisms of groups, the classification of cyclic groups; examples — the sign of a permutation, inversion in an abelian group, the exponential, the determinant, $S_3$ and $D_3$; subgroups, kernel and image, the injectivity criterion; left and right cosets and the partition into them; normal subgroups and conjugation, the index, Lagrange's theorem, subgroups of index 2; the quotient group; the homomorphism theorem with examples; conjugate elements, the centre, a group action on a set.