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(a) Prove that |GLn(F,)| = IIq" – ' -!). j=1 Hint: Do this one column at a time; explain why there are q" – 1 choices for the first column, q" – q choices for the second column (think linear independence), and so on. | (b) Deduce that |GL,(Z»)| = p"°(k=1) I[P" – p°-1). j=1 Hint: Show that : Mn(Zpk) → Mn(Zp) by (A) = A mod p is a surjective ring homomorphism, apply the First Isomorphism Theorem, and finally restrict to the group of invertible matrices. %3D

(a) Prove that |GLn(F,)| = IIq" – ' -!). j=1 Hint: Do this one column at a time; explain why there are q" – 1 choices for the first column, q" – q choices for the second column (think linear independence), and so on. | (b) Deduce that |GL,(Z»)| = p"°(k=1) I[P" – p°-1). j=1 Hint: Show that : Mn(Zpk) → Mn(Zp) by (A) = A mod p is a surjective ring homomorphism, apply the First Isomorphism Theorem, and finally restrict to the group of invertible matrices. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Prove that |GLn(F,)| = II" – q'-!). j=1 Hint: Do this one column at a time; explain why there are q" – 1 choices for the first column, q" - q choices for the second column (think linear independence), and so on. (b) Deduce that |GL„(Z„+)| = pr* (k=1) II»" – p¯-1). j=1 Hint: Show that v : Mn(Zpk) → Mn(Zp) by (A) = A mod p is a surjective ring homomorphism, apply the First Isomorphism Theorem, and finally restrict to the group of invertible matrices.
Transcribed Image Text:(a) Prove that |GLn(F,)| = II" – q'-!). j=1 Hint: Do this one column at a time; explain why there are q" – 1 choices for the first column, q" - q choices for the second column (think linear independence), and so on. (b) Deduce that |GL„(Z„+)| = pr* (k=1) II»" – p¯-1). j=1 Hint: Show that v : Mn(Zpk) → Mn(Zp) by (A) = A mod p is a surjective ring homomorphism, apply the First Isomorphism Theorem, and finally restrict to the group of invertible matrices.
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