Groups A and B have the same order and have the same number of subgroups, are they necessarily isomorphic? If this is true, can I use it in an A-Level MEI FP3 answer without explaining it?
I came across a past paper question asking me to explain why all groups of order 5 are always isomorphic to eachother. The first thing I wrote down was "5 is a prime number so by Lagrange's Theorem, the only subgroups it can have is itself and a group only containing its identity"
So the second part of my answer is "So all order 5 groups have the same number of subgroups, therefore all of them are isomorphic to eachother". But I am not sure if this is correct.