1. Let G be a group and H a nonempty subset of G. Then H

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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1. Let G be a group and H a nonempty subset of G. Then H <G if
abEH whenever a,bƐH
2. Let G be a group and H a subgroup of G. Show that C(H) <G.
3. If H. :aEA are a family of subgroups of the group G, show that
is a subgroup of G.
4. Let H - {a + bi | a,bɛR, ab z 0} . Determine whether H is a subgroup of the
complex numbers C with addition.
5. Let a be an element of order n in a group and let k be a positive
integer. Then <a* >=< a™d mA)>
6. Let a be an element of order n in a group and let k be a positive
integer. Then l- gcd(n,.k)
7. Every subgroup of a cyclic group is cyclic.
8. Determine the subgroup lattice of Z
9. Prove that a group of order 3 must be cyclic.
la*| =
Transcribed Image Text:1. Let G be a group and H a nonempty subset of G. Then H <G if abEH whenever a,bƐH 2. Let G be a group and H a subgroup of G. Show that C(H) <G. 3. If H. :aEA are a family of subgroups of the group G, show that is a subgroup of G. 4. Let H - {a + bi | a,bɛR, ab z 0} . Determine whether H is a subgroup of the complex numbers C with addition. 5. Let a be an element of order n in a group and let k be a positive integer. Then <a* >=< a™d mA)> 6. Let a be an element of order n in a group and let k be a positive integer. Then l- gcd(n,.k) 7. Every subgroup of a cyclic group is cyclic. 8. Determine the subgroup lattice of Z 9. Prove that a group of order 3 must be cyclic. la*| =
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