S4 is a subgroup of S4, of order 24. Its Cayley table is 12 elements 8 elements 6 elements 4 elements 3 elements 2 elements Each element of order 2 in S4 generates a subgroup } of order 2. There are 9 such elements: the transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23). 1 element The trivial subgroup is the unique subgroup of order 1 in S4. Other examples The even integers form a subgroup 2Z of the integer ring Z: the sum of two even integers is even, and the negative of an even integer is even. An ideal in a ring is a subgroup of the additive group of . A linear subspace of a vector space is a subgroup of the additive group of vectors. In an abelian group, the elements of finite order form a subgroup called the torsion subgroup. See also Cartan subgroup Fitting subgroup Fixed-point subgroup Fully normalized subgroup Stable subgroup Notes References . . . Group theory Subgroup properties