GATE 2026 MA – Question 54

Algebra · Groups · 2 marks · Multiple select

Let G=ℤ₃₀⊕ℤ₁₂ under addition. Which statements hold?

G is not cyclic.
The order of (14,7) is 98.
The order of (14,7) is 60.
G has an element of order 360.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) G is not cyclic.; (C) The order of (14,7) is 60.

Explanation

Component orders of (14,7) are $30/\gcd(30,14)=15$ and $12/\gcd(12,7)=12$. Pair order is lcm(15,12)=60, proving C. Every element order divides lcm(30,12)=60, less than |G|=360, so no generator or order-360 element exists. **A,C**.

More Groups questions

GATE 2025 MA Q64 – In a group G with identity e, g≠e, g²=e, h≠e, h²≠e and ghg⁻¹=h². Find the least positive n…
GATE 2024 MA Q16 – For a group of order 2024, n p is its Sylow p-subgroup count. Which statement always…
GATE 2024 MA Q27 – How many nonisomorphic finite groups have exactly three conjugacy classes? (integer)
GATE 2024 MA Q31 – How many group homomorphisms ℤ/4ℤ→S₄ exist? (integer)
GATE 2024 MA Q40 – I: Some proper subgroup G of (ℚ,+) has finite quotient ℚ/G. II: Some subgroup G has…
GATE 2023 MA Q36 – In GL₂(ℂ), let S=pmatrix0&-1\\1&0pmatrix, T₁=iI,T₂=diag(i,−i), T 3=pmatrix0&1\\1&0pmatrix…

All Groups questions →