Quiz:Sylow number

{What are the various possibilities for the number of 3-Sylow subgroups in a group of order 72? - 1 or 2 - 1 or 3 + 1 or 4 - 1, 2, 4, or 8 - 1, 3, or 9
 * type=""}
 * Follows by combining Sylow number equals index of Sylow normalizer and congruence condition on Sylow numbers. See subgroup structure of groups of order 72.

{Which of the following combinations for the number of 2-Sylow subgroups and the number of 3-Sylow subgroups is not possible in a group of order 24? - 1 2-Sylow subgroup, 1 3-Sylow subgroup - 1 2-Sylow subgroup, 4 3-Sylow subgroups - 3 2-Sylow subgroups, 1 3-Sylow subgroup - 3 2-Sylow subgroups, 4 3-Sylow subgroups + None of the above, i.e., they are all possible
 * type=""}
 * See subgroup structure of groups of order 24.

{Which of the following combinations for the number of 2-Sylow subgroups and the number of 7-Sylow subgroups is not possible in a group of order 56? - 1 2-Sylow subgroup, 1 7-Sylow subgroup - 1 2-Sylow subgroup, 8 7-Sylow subgroups - 7 2-Sylow subgroups, 1 7-Sylow subgroup + 7 2-Sylow subgroups, 8 7-Sylow subgroups - None of the above, i.e., they are all possible.
 * type=""}
 * Follows by size considerations -- there isn't enough space to accommodate all the subgroups. See order is product of Mersenne prime and one more implies normal Sylow subgroup. The Mersenne prime here is $$7 = 2^3 - 1$$. See also subgroup structure of groups of order 56.