Question

The group $Z_4$ under addition modulo $4$ has

• A. Exactly one proper subgroup
• B. Only two proper subgroups
• C. No subgroups
• D. Infinitely many proper subgroups

Answer 1

Answer: (B)

Only two proper subgroups

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

Let $H_1=\{0,2\}\subset Z_4$ and $H_2=\{0,1,3\}\subset Z_4$

Identity $0\in H_1$

$2+2=4=0⟹2^{-1}=2$

$\therefore$ inverse exist for every element of $H_1$ and also, closure property is satisfied as $0+2\in H_1$

Thus, $H_1$ is a proper subgroup of $(Z_4,+)$

Similarly,

Identity $0\in H_2$

$1+3=4=0⟹1$ and $3$ are inverse of each other and they belong to $H_2$

$\therefore$ inverse exist for every element of $H_2$ and also, closure property is satisfied as $1+3=0,0+3=3,0+1=1\in H_2$

Thus, $H_2$ is a proper subgroup of $(Z_4,+)$

Hence, $Z_4$ has only two proper subgroups.