MAT 3542 [Group Theory] Tut 4 2023 (a) Let G be an abelian group. Show that the elements of finite order in G form a subgroups This subgroup is called the torsion subgroup of G. (b) Let G be a group. If : ZxZ-G is a homomorphism and D(1,0) - h while D(0, 1)k, find (m,n). (c) Prove that the torsion subgroup H of an abelian group G is a normal subgroup of G, a that G/H is torsion free. An abelian group is torsion free ife is the only element of finite of (d) Show that if H and N are subgroups of a group G, and N is a normal subgroup, then is normal in H. F (e) Show that the set of all ge G such that ig: GG is the identity inner automorphi is a normal subgroup of a group G. (f) Let F be a multiplicative group of all continuous functions mapping R into R that a nonzero at every x e R. Let R* be the multiplicative group of nonzero real numbers. Le * be given by ( (fr)dy Is a homomorphism?
MAT 3542 [Group Theory] Tut 4 2023 (a) Let G be an abelian group. Show that the elements of finite order in G form a subgroups This subgroup is called the torsion subgroup of G. (b) Let G be a group. If : ZxZ-G is a homomorphism and D(1,0) - h while D(0, 1)k, find (m,n). (c) Prove that the torsion subgroup H of an abelian group G is a normal subgroup of G, a that G/H is torsion free. An abelian group is torsion free ife is the only element of finite of (d) Show that if H and N are subgroups of a group G, and N is a normal subgroup, then is normal in H. F (e) Show that the set of all ge G such that ig: GG is the identity inner automorphi is a normal subgroup of a group G. (f) Let F be a multiplicative group of all continuous functions mapping R into R that a nonzero at every x e R. Let R* be the multiplicative group of nonzero real numbers. Le * be given by ( (fr)dy Is a homomorphism?
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 5E: 5. Exercise of section shows that is a group under multiplication.
a. List the elements of the...
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![University of Venda
MAT 3542 [Group Theory] Tut 4 2023
(a) Let G be an abelian group. Show that the elements of finite order in G form a subgroup.
This subgroup is called the torsion subgroup of G.
(b) Let G be a group. If
D(0, 1)k, find (m,n).
: ZxZ- G is a homomorphism and D(1,0) while
(c) Prove that the torsion subgroup H of an abelian group G is a normal subgroup of G, and
that G/H is torsion free. An abelian group is torsion free ife is the only element of finite order.
(d) Show that if H and N are subgroups of a group G, and N is a normal subgroup, then HN
is normal in H.
(e) Show that the set of all g e G such that ig: GG is the identity inner automorphism, i..
is a normal subgroup of a group G.
(f) Let F be a multiplicative group of all continuous functions mapping R into R that are
nonzero at every xe R. Let R* be the multiplicative group of nonzero real numbers. Let :
F→ R* be given by D() = f(x)dx. Is a homomorphism?
(g) Let G be any group let a be any element of G. Let : Z→G be defined by Þ(n) = a".
Show that is a homomorphism.
(h) Let G be a group. Let h, k eG and let : ZxZ→G be defined by D(m, n) =h™k".
Give a necessary and sufficient condition, involving h and k, for be a homomorphism. Prove
your condition.
cientific WorkPlace
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Transcribed Image Text:University of Venda
MAT 3542 [Group Theory] Tut 4 2023
(a) Let G be an abelian group. Show that the elements of finite order in G form a subgroup.
This subgroup is called the torsion subgroup of G.
(b) Let G be a group. If
D(0, 1)k, find (m,n).
: ZxZ- G is a homomorphism and D(1,0) while
(c) Prove that the torsion subgroup H of an abelian group G is a normal subgroup of G, and
that G/H is torsion free. An abelian group is torsion free ife is the only element of finite order.
(d) Show that if H and N are subgroups of a group G, and N is a normal subgroup, then HN
is normal in H.
(e) Show that the set of all g e G such that ig: GG is the identity inner automorphism, i..
is a normal subgroup of a group G.
(f) Let F be a multiplicative group of all continuous functions mapping R into R that are
nonzero at every xe R. Let R* be the multiplicative group of nonzero real numbers. Let :
F→ R* be given by D() = f(x)dx. Is a homomorphism?
(g) Let G be any group let a be any element of G. Let : Z→G be defined by Þ(n) = a".
Show that is a homomorphism.
(h) Let G be a group. Let h, k eG and let : ZxZ→G be defined by D(m, n) =h™k".
Give a necessary and sufficient condition, involving h and k, for be a homomorphism. Prove
your condition.
cientific WorkPlace
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