Answer to Question #177192 in Abstract Algebra for bhavana

Question #177192

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


1
Expert's answer
2021-04-14T14:44:06-0400

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"

 

Identity 0∈"H _1"

​  

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"

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

Similarly, 

Identity 0∈"H _2"

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

 ∴ 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"

 

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.


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