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Leg G be a finite group and let H be a subgroup of G. Then the order of H is a factor of the order of G. More precisely, |G|=m|H| where m is the number of different cosets of H in G.

Can someone clarify what the | | means?

I thought it was how many elements are in a group, such as, the symmetric group of 4 (S4) has 4! elements, so |S4| = 24.

I have an example saying the S4 cannot have a subgroup of order 5 since |S4| = 24 which is not an exact multiple of 5.

But 24 is the number of elements in S4, not the order of the group.. So why are they saying the subgroup can't have order 5 because of the number of elements in S4? Surely we should be finding the order of S4 instead to se if there is a subgroup of order 5 in the group.

Thanks