GATE 2020 CS Q.18: solution and answer

GATE 2020 · CSQ.18 · 1 mark · NAT

Let $G$ be a group of 35 elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is ______.

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Answer and solution

ANSWER

The largest possible size of a proper subgroup of a group of 35 elements is 7.

Official answer: 7Source: Official answer key, GATE 2020 CS (IIT Delhi)

By Lagrange's theorem, the order of every subgroup of a finite group divides the order of the group.
So a subgroup of $G$ has order dividing $35 = 5 \times 7$ , that is, one of $1, 5, 7, 35$ .
Excluding $G$ itself (order $35$ ), the possible orders are $1, 5, 7$ .
The largest of these is $7$ . It is attained: since $7$ is a prime dividing $∣ G ∣ = 35$ , Cauchy's theorem gives an element of order $7$ . It generates a cyclic subgroup of order $7$ , which is a proper subgroup of $G$ .
Hence the largest possible size of a proper subgroup of $G$ is $7$ .

KEY IDEA

Lagrange's theorem restricts a subgroup's order to a divisor of 35. Cauchy's theorem shows the largest proper divisor, 7, is actually achieved.

WATCH OUT FOR

Stopping at Lagrange's theorem and taking the largest proper divisor without checking that a subgroup of that order exists. Divisibility is only necessary, and the existence of the order-7 subgroup comes from Cauchy's theorem.

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GATE 2020 CS Q.18: solution and answer

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