1-2 page math(abstract algebra) note summary

Jing Cao

In class 3, the properties of the identity and the inverse of a group were

discussed. The identity and the inverse of a group are unique and (a−1)−1 = a

and (ab)−1 = b−1a−1 for any a, b ∈ G. By cancellation law, au = av ⇒ u = v

and ua = va ⇒ u = v. For any a1, a2, ...an, the product a1 ·a2... ·an is indepen-

dent of how the bracket is expressed. A symmetric group is the group whose

elements are all the bijections from a set to itself, and the elements of a sym-

metric group are the permutations of {1, 2, ...n}. We can write a permutation

in cycle notation. The order of a permutation is the least common multiple of

the lengths of the disjoint cycles when the permutation is written as a product

of disjoint cycles.

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notes/Class4.pdf