Question

verify whether (z4,+4) is a group z4 is the set of integers modulo 4 and +4 is addition modulo 4

Answer 1

Only two proper subgroups

$Z_4 =$ {$0,1,2,3$ }

Let $H _1$ ={0,2}⊂$Z_ 4$ and $H _2$ ={0,1,3}⊂$Z_ 4$

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Identity 0∈$H _1$

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2+2=4=0⟹2 ^{−1} =2

∴ inverse exist for every element of $H _1$ and also, closure property is satisfied as 0+2∈$H _1$

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 Thus, $H _1$ is a proper subgroup of $(Z _4,+4)$

Similarly, 

Identity 0∈$H _2$

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1+3=4=0⟹1 and 3 are inverse of each other and they belong to $H _2$

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 ∴ inverse exist for every element of $H _2$ and also, closure property is satisfied as 1+3=0,0+3=3,0+1=1∈$H _2$

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Thus, $H _2$ is a proper subgroup of $(Z _4,+4)$

Hence $,(Z_4,+4)$ is a group $Z_4$ is the set of integers modulo 4.