Simple groups of order 60 are important in group theory because they are the smallest non-abelian simple groups. They also have interesting properties and connections to other mathematical concepts.
To prove this statement, we first need to show that any simple group of order 60 must have a normal subgroup of order 12. This can be done using the Sylow theorems. Then, we show that this normal subgroup is isomorphic to A5 by constructing a specific isomorphism between the two groups.
This is because A5 is the only group of order 60 that has a normal subgroup of order 12. Any other group of order 60 would have a different normal subgroup, and therefore would not be isomorphic to A5.
One example of a simple group of order 60 is the alternating group A5, which consists of all even permutations on 5 elements. This group has order 60 and is isomorphic to A5.
Yes, there are other ways to prove this statement. One approach is to use the classification of finite simple groups, which states that any simple group of order less than 60 must be cyclic or alternating. Another approach is to use representation theory and show that any simple group of order 60 must have a faithful irreducible 5-dimensional representation, which is unique up to isomorphism and is equivalent to A5.