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One of the basic tenents of cube theory (a highly specialized branch of group theory - grin!) is that odd permutations of the corners can only occur with odd permutations of the edges, and that even permutations of the corners can only occur with even permutations of the edges. This fact is one of the three factors that causes the order of the cube group to be smaller than the order of what is sometimes the illegal cube group.

If you take a cube apart and put it back together, you can put it back together in twelve times as many ways as there are positions in the cube group. Twelve is because 3 * 2 * 2 = 12. Three of the ways are because the twist of the last corner cubie you put back together can be set in one of three different ways, only one of which is legal. Two of the ways are because the flip of the last edge cubie you put back together can be set in one of two different ways, only one of which is legal. The last two of the ways are because the corners and edges can by put back together either with the same odd/even parity (legal) or with the opposite odd/even parity (illegal).