← 2015 Paper 2

UPSC 2015 Maths Optional Paper 2 Q1a-i — Step-by-Step Solution

5 marks · Section A

Cyclic groups · Algebra · asked 8× in 13 yrs · Read the full method →

Question

How many generators are there of the cyclic group $G$ of order 8? Explain.

Technique

Euler’s totient $\varphi(8)$ counts the generators of $\mathbb Z_8$ (equivalently any cyclic group of order 8).

Solution

Key fact. A cyclic group of order $n$, generated by $a$, has element $a^k$ as a generator iff $\gcd(k,n)=1$. The number of generators is $\varphi(n)$ (Euler’s totient).

Step 1 — Apply to $n=8$

Test each $k$:

• $\gcd(1,8)=1$ ✓
• $\gcd(2,8)=2$ ✗
• $\gcd(3,8)=1$ ✓
• $\gcd(4,8)=4$ ✗
• $\gcd(5,8)=1$ ✓
• $\gcd(6,8)=2$ ✗
• $\gcd(7,8)=1$ ✓

Four values coprime to 8: $k\in\{1,3,5,7\}$.

Step 2 — Conclusion

Answer