Let T be the group of nonsingular upper triangular 2 × 2 matrices with entries in R; that is, matrices of the form 4. where a, b, c E R and ac # 0. Let U consist of matrices of the form where a € R. (a) Show that U is a subgroup of T. (b) Prove that U is abelian. (c) Prove that U is normal in T. (d) Show that T/U is abelian. (e) Is T normal in GL2(R)?
Let T be the group of nonsingular upper triangular 2 × 2 matrices with entries in R; that is, matrices of the form 4. where a, b, c E R and ac # 0. Let U consist of matrices of the form where a € R. (a) Show that U is a subgroup of T. (b) Prove that U is abelian. (c) Prove that U is normal in T. (d) Show that T/U is abelian. (e) Is T normal in GL2(R)?
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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Just 4e
Transcribed Image Text:Let T be the group of nonsingular upper triangular 2 × 2 matrices with entries in R;
that is, matrices of the form
4.
(: )
where a, b, c E R and ac # 0. Let U consist of matrices of the form
(6 ).
where a € R.
(a) Show that U is a subgroup of T.
(b) Prove that U is abelian.
(c) Prove that U is normal in T.
(d) Show that T/U is abelian.
(e) Is T normal in GL2(R)?
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