Maths cafe

 Cauchy theorem for finite abelian groups :  If G is a finite abelian group such that p/ o( G) , p is a prime number , then there exists an element a( ≠ e) ∈ G such that  a^p= e  i.e. o(a) = p. Cauchy theorem for finite groups :  If G is a finite group such that p/ o (G)  , p is a prime number , then  there exists an element of order p in G.

 Let 𝐺 =< 𝑎, 𝑏: 𝑎^3 = 𝑏^2 = (𝑎𝑏)^2 = 1 > be the dihedral group of order 8. Its  elements are {1, 𝑎, 𝑎^2,  𝑏, 𝑎𝑏, 𝑎^2𝑏, }.  Subgroups of D3: <1>={1}, <a>={a , a^2,a^3=1} <a^2>={a^2,a^4=a, a^6=1} <b>={b, b^2=1} <ab>={ab, 1} <a^2b>={a^2b, 1} Center of D3: Z(D3)={e}.               because n is odd Commutator subgroup of D3: {1,a, a^2} Normal subgroup of D3: t(3)+1=2+1=3 Cyclic subgroup of D3: t(3)+3=2+3=5 D3 smallest non-ablian group