Let M denote the group of 2 x 2 non-singular real matrices under multiplication. Let O denote the subgroup of orthogonal matrices (i.e., A¬1 = A") in M. a. Show O decomposes as the disjoint union of 0+ = { A € 0 \Det(A) = 1} and O= = { A e 0 \Det(A) = –1}. b. Is O+ a normal subgroup of 0? Is O¯? c. Is O+ a normal subgroup of M?

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let M denote the group of 2 x 2 non-singular real matrices under multiplication.
Let O denote the subgroup of orthogonal matrices (i.e., A-1 = A") in M.
a. Show O decomposes as the disjoint union of 0+ = { A e 0 |Det(A) = 1} and
0- = { A e 0 ||Det(A) = –1}.
b. Is O+ a normal subgroup of O? Is 0¯?
c. Is O+ a normal subgroup of M?
Transcribed Image Text:Let M denote the group of 2 x 2 non-singular real matrices under multiplication. Let O denote the subgroup of orthogonal matrices (i.e., A-1 = A") in M. a. Show O decomposes as the disjoint union of 0+ = { A e 0 |Det(A) = 1} and 0- = { A e 0 ||Det(A) = –1}. b. Is O+ a normal subgroup of O? Is 0¯? c. Is O+ a normal subgroup of M?
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