Is there a simple group of order 60?
Here we show that 60 is the smallest composite order of a noncommutative simple group and any noncommutative simple group of order 60 is isomorphic to A5. It follows that n5 = 1, that is, G has unique sylow 5-subgroup H of order 5. Hence H is normal in G, and so G is not simple.
What are the possible orders for non Abelian simple groups?
There are only 491 possible orders for non-Abelian simple groups of order less than 10 billion.
How many Sylow 5 subgroups does G have?
G has six Sylow -5 subgroups. G has four Sylow -3 subgroups.
Is group of order 60 Abelian?
60 is the smallest possible order of a simple non-abelian group.
How many generators does a cyclic group of order 60 have?
16 Generators
2 Answers. No of generators in Group (Cyclic group) too is given by Euler’s_totient_function, i.e. no of elements less than N & Co prime to N. No of generators possible are =60(1−1/2)(1−1/3)(1−1/5)=60∗1/2∗2/3∗4/5=16. So total 16 Generators !
What is the order of a group?
The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.
Are there infinite simple groups?
Classification. There is as yet no known classification for general (infinite) simple groups, and no such classification is expected.
How many Sylow 5 subgroups does S5 have?
6 Sylow
S5: 120 elements, 6 Sylow 5-subgroups, 10 Sylow 3-subgroups, and 15 Sylow 2-subgroups. A typical Sylow 5-subgroups is {e,(12345),(13524),(14253),(15432)}, which has normalizer 〈(12345),(2354)〉 with order 20.
How many Sylow 5 subgroups of S5 are there here S5 is the symmetric group?
six Sylow-5 subgroups
Now, (32145)(12)(54123)=(14), and (14)(12345)(14)=(42315). Spring 2008 Problem 1. The symmetric group S5 has six Sylow-5 subgroups. This implies that S6 contains two copies of S5 that are not conjugate to each other.