Mathematics · Group Theory
Let SL2(ℤ) = {( a b
c d ) ∈ M2(ℤ) : ad - bc = 1} and for any prime p, let the principal congruence subgroup be defined as:
Γ(p) = {( a b
c d ) ∈ SL2(ℤ) | a ≡ 1 (mod p), d ≡ 1 (mod p), c ≡ 0 (mod p), b ≡ 0 (mod p)}.
Which of the following statements is true?
c d ) ∈ M2(ℤ) : ad - bc = 1} and for any prime p, let the principal congruence subgroup be defined as:
Γ(p) = {( a b
c d ) ∈ SL2(ℤ) | a ≡ 1 (mod p), d ≡ 1 (mod p), c ≡ 0 (mod p), b ≡ 0 (mod p)}.
Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1Γ(p) is a finite group
- 2Γ(p) is uncountable
- 3Γ(p) is not a subgroup of SL 2 (ℤ)
- 4Γ(p) is a normal subgroup of SL 2 (ℤ)✓
Correct answer
Γ(p) is a normal subgroup of SL 2 (ℤ)
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