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D3

 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  


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