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The multiplicative group of Z_{30} has 8 elements (\phi(30). {1,7,11,13,17,19,23,29}. Here are the 8 subgroups {1,11}, {1,7,13, 19}, {1,17,19,23}, {1, 19}, {1,17,19,23}, {1,29}, {1}, and the whole thing.

The group of part 1 D_4 (some authors call it D_8). The symmetries of the square. i.e. 1-1 and onto mappings from the square to itself. It is a subgroup of S_4 generated by the cycle (1234) and the transposition (13)

Volterra To begin with you need to find a function of two variables (x_1, x_2) which is a constant of motion. i.e. its time derivative is zero. Then you worry where it is defined.