32. If A = | is a matrix, the number ad = be is denoted det A and called the determinant of A. Prove that the function f:GL(2, R)→R* given by S(A) = det A is a surjective homomorphism.
32. If A = | is a matrix, the number ad = be is denoted det A and called the determinant of A. Prove that the function f:GL(2, R)→R* given by S(A) = det A is a surjective homomorphism.
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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b My Questions | bartleby
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\fla)) = la].
Capyright 2012 Cmlanng A1 Righo Ramud May aot be opA cnd ar datikndin whale aria pan Due to earanic de,ne rd party cantet any deppnd nte Boak adkarear). Rrial vi e
d d et aog ed co du ot miy deca tbe ove o eina Ca Laingaam rige o adinal co t eg time it og kainque it.
7.4 Isomorphisms and Homomorphisms 225
30. Let f:G→ H be a homomorphism of groups and let K be a subgroup of H.
Prove that the set {aɛG|f(a)EK} is a subgroup of G.
31. If f:G→ Gis a homomorphism of groups, prove that F= {aEG|f(@) = a} is
a subgroup of G.
(:)
32. If A =
is a matrix, the number ad - be is denoted det A and called
the determinant of A. Prove that the function f:GL(2, R)→R* given by
= det A is a surjective homomorphism.
33. Let f:G→ H be a homomorphism of groups and let K,= {aeG|f(a) = en},
that is, the set of elements of G that are mapped by fto the identity element
of H. Prove that K, is a subgroup of G. See Exercises 34 and 35 for examples.
34. The function f.Z→Z, given by f(x) = [x] is a homomorphism by Example 13.
Find K(notation as in Exercise 33).
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Transcribed Image Text:Thomas W. Hungerford - Abstrac x
b My Questions | bartleby
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\fla)) = la].
Capyright 2012 Cmlanng A1 Righo Ramud May aot be opA cnd ar datikndin whale aria pan Due to earanic de,ne rd party cantet any deppnd nte Boak adkarear). Rrial vi e
d d et aog ed co du ot miy deca tbe ove o eina Ca Laingaam rige o adinal co t eg time it og kainque it.
7.4 Isomorphisms and Homomorphisms 225
30. Let f:G→ H be a homomorphism of groups and let K be a subgroup of H.
Prove that the set {aɛG|f(a)EK} is a subgroup of G.
31. If f:G→ Gis a homomorphism of groups, prove that F= {aEG|f(@) = a} is
a subgroup of G.
(:)
32. If A =
is a matrix, the number ad - be is denoted det A and called
the determinant of A. Prove that the function f:GL(2, R)→R* given by
= det A is a surjective homomorphism.
33. Let f:G→ H be a homomorphism of groups and let K,= {aeG|f(a) = en},
that is, the set of elements of G that are mapped by fto the identity element
of H. Prove that K, is a subgroup of G. See Exercises 34 and 35 for examples.
34. The function f.Z→Z, given by f(x) = [x] is a homomorphism by Example 13.
Find K(notation as in Exercise 33).
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