short_exact_sequence

def: short exact sequence of groups A, B, C is a sequence 0 --> A --> B --> C --> 0 with f: A --> B injective, g: B --> C surjective, and ker g = im f. such a sequence splits if there exists h: C --> B such that hg = id_C, or equivalently, B = A  (+) C.

ex. consider the movements of the chess pieces, except pawns, under the group operation of composition of moves (and assume a piece can choose not to move, so there exists an identity element). let A = bishop, B = queen, C = rook. then 0 --> A --> B --> C --> 0 is a short exact sequence. however, such a sequence does not split because the queen is not exactly bishop + rook.