Let p be a prime number. Let G be a group such that for each g ∈ G there exists an n ∈ ℕ such that g^pⁿ = 1. Which of the following statements is FALSE?

Question

Question

This question was previously asked in

CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)

Answer 1

Concept:

(i) Sylow’s First Theorem: Let G be a finite group and let p be a prime. If p^k divides |G|, then G has at least one subgroup of order p^k.

(ii) If f o(G) = pn then there exist at least n - 1 normal subgroup of order pr for all 0 ≤ r ≤ n

Explanation:

(1): Given |G| = p6, so p4 divides p6. Then by Sylow's first theorem, there exist a subgroup H ⊂ G such that |H| = p4

Therefore index of H = p6/ p4 = p2

Option (1) is TRUE

(2): |G| = p6 then by result (ii) G has at least five normal subgroups

Option (2) is TRUE and option (4) is FALSE

(3): If we consider (P) and p = 2 then center of G is infinite.

Option (3) is TRUE

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