I am answering a question regarding the simplicity of a group of order 380 and wondering if my approach is okay.
380 = 2^2 * 5 * 19, so the Sylow 5 subgroups have order 5, are cyclic, and if there are more than 1 of them, then we must have 76 of them. Hence, we have (5-1)*76=304 elements of order 5. Now, looking at the Sylow 19 subgroups, if there are greater than 1 of them there must be 20, but we don’t have enough space for all of those elements in the group, hence there is only one Sylow 19 subgroup, a normal subgroup, and so the group is not simple.
