Klein's four group

 Def. Klein’s four group :

A group of order four in which every element is self-inverse or every non-identity 

element is of order 2 is called Klein’s four group. Symbolically, V4={a, b, c, d} such that

ab=ba=c

bc=cb=a

ca=ac=b

and a^2=b^2=c^2 =e

Properties:

a) Every non-identity element is of order 2.

b) Any two of the three non-identity element generates the third one.

c) It is the smallest non-cyclic group.

d) All proper subgroups of V4 are cyclic.

Subgroups of V4:

H1=e

𝐻2 ={ 𝑒, 𝑎} , 

𝐻4 ={ 𝑒, 𝑐 }

 𝐻3 = {𝑒, 𝑏},

 𝐻5 = 𝐾4