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UPSC 2022 Maths Optional Paper 2 Q1a — Step-by-Step Solution

10 marks · Section A

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

Question

Show that the multiplicative group $G=\{1,-1,i,-i\}$ is isomorphic to $G'=(\{0,1,2,3\},+_4)$.

Technique

Both groups are cyclic of order 4 (both generated by an element of order 4); the isomorphism maps generator to generator.

Solution

Strategy. Both groups have order 4 and are cyclic. Identify a generator for each, then map generator to generator.

Step 1 — Verify both are cyclic of order 4

$G$ under multiplication: $i^0=1,\,i^1=i,\,i^2=-1,\,i^3=-i,\,i^4=1$. So $G=\langle i\rangle$, cyclic of order 4.

$G'$ under $+_4$: $0,1,2,3$ with addition mod 4. Generator is $1$: $1,1+1=2,2+1=3,3+1=0$. $G'=\langle 1\rangle$, cyclic of order 4.

Step 2 — Define isomorphism

Let $\phi:G\to G'$ by $\phi(i^k)=k\pmod 4$, i.e.:

• $\phi(1)=0$
• $\phi(i)=1$
• $\phi(-1)=2$
• $\phi(-i)=3$

Step 3 — Verify $\phi$ is a homomorphism

For $i^a,i^b\in G$: $\phi(i^a\cdot i^b)=\phi(i^{a+b})=(a+b)\bmod 4=\phi(i^a)+_4\phi(i^b)$.

Sample check: $\phi(i\cdot(-1))=\phi(-i)=3$. And $\phi(i)+_4\phi(-1)=1+_4 2=3$ ✓.

Step 4 — Verify $\phi$ is bijective

$\phi$ is clearly a bijection (one-to-one correspondence between the 4 elements).

Answer