Exercise 970Algebra › GroupsL1Added 3 Nov 2026

Counting cosets

#cosets#index#counting

Statement

In $G=\mathbb{Z}/12\mathbb{Z}$ , let $H=\langle\bar4\rangle$ . Compute $|H|$ , the index $[G:H]$ , and list the elements of $H$ and $G/H$ .

Hint

$\bar4$ has order $12/\gcd(4,12)=3$ .

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Solution

$|H|$ . Order of $\bar4$ is $12/4=3$ . So $H=\{\bar0,\bar4,\bar8\}$ .

Index. $[G:H]=12/3=4$ .

Cosets. $G/H$ has $4$ elements:
- $H=\{\bar0,\bar4,\bar8\}$ ;
- $\bar1+H=\{\bar1,\bar5,\bar9\}$ ;
- $\bar2+H=\{\bar2,\bar6,\bar{10}\}$ ;
- $\bar3+H=\{\bar3,\bar7,\bar{11}\}$ .

Conclusion. $|H|=3$ , $[G:H]=4$ , and $G/H\cong\mathbb{Z}/4\mathbb{Z}$ .

Counting cosets

Statement

Solution

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