The GATE Grind

GATE 2026 MA – Question 14

Algebra · Groups · 1 mark · Multiple choice

A group G has order 595. Which statement is true?

  1. G cannot have a proper normal subgroup.
  2. G must have a proper normal subgroup.
  3. G cannot have an element of order 17.
  4. G has 17 Sylow 5-subgroups.

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Correct answer: (B) G must have a proper normal subgroup.

Explanation

Factor 595 = 5×7×17. The Sylow 5-subgroup count divides 119, so it is one of 1,7,17,119, and must be 1 modulo 5. Only 1 qualifies. The unique Sylow 5-subgroup is a nontrivial proper normal subgroup. **B**. Cauchy’s theorem also guarantees an element of order 17.

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