5. Suppose [G: H] = n. Show that g" E H for all g € G. Recall: [G: H] is the number of (left) cosets of H, which is also the size of G/H.
5. Suppose [G: H] = n. Show that g" E H for all g € G. Recall: [G: H] is the number of (left) cosets of H, which is also the size of G/H.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Number 5
![H
ts:
still cap
ed with another element that com-
with h. (Also see Chapter 5, Exer-
(in either order) is an element that doesn't commute with h. This is captured by
of D4 that commutes with h composed with an element that doesn't
EHF90H = rooH EH = 90H.
. An element of D₁ that doesn't commute with h composed with another element
that doesn't is an element that commutes with h. This is captured by r90H. r90H =
EH.
(aH-bH)-1¹ =
2. Consider the following statement:
Exercises
Unless specified otherwise, assume that the coset multiplication shortcut holds in G/H.
1. Since a quotient group is a group, any property that we know about groups applies
to G/H as well. Let's consider the "socks-shoes," for example. What goes into the
empty boxes?
H
If aH=bH in G/H, then a = b in G.
Is it true or false? If it's true, prove it. If it's false, give a counterexample.
3. Let H = {1, 3, 9} be a subgroup of U13, and consider U₁3/H = {1H, 2H, 4H, 7H}.
(a) Find all integers n for which (2H)n = 1H in U13/H.
(b) Find all integers n for which 2" E H.
(c) What conjecture do you have?
4. Prove: Let gH E G/H and n E Z. Then (gH)n = H if and only if g" EH.
5. Suppose [G: H] = n. Show that g" E H for all g € G.
Recall: [G: H] is the number of (left) cosets of H, which is also the size of G/H.
9.
i.e.,
#19.
Hint
inclu
10. Is th
als
11. Cc
ab
12. L
2
13.
14.
15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47675d24-ba7f-4e21-93eb-97faaaed8f8d%2F4b95c95b-26d1-432c-929a-c11b1d28e040%2Fz50xz0l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:H
ts:
still cap
ed with another element that com-
with h. (Also see Chapter 5, Exer-
(in either order) is an element that doesn't commute with h. This is captured by
of D4 that commutes with h composed with an element that doesn't
EHF90H = rooH EH = 90H.
. An element of D₁ that doesn't commute with h composed with another element
that doesn't is an element that commutes with h. This is captured by r90H. r90H =
EH.
(aH-bH)-1¹ =
2. Consider the following statement:
Exercises
Unless specified otherwise, assume that the coset multiplication shortcut holds in G/H.
1. Since a quotient group is a group, any property that we know about groups applies
to G/H as well. Let's consider the "socks-shoes," for example. What goes into the
empty boxes?
H
If aH=bH in G/H, then a = b in G.
Is it true or false? If it's true, prove it. If it's false, give a counterexample.
3. Let H = {1, 3, 9} be a subgroup of U13, and consider U₁3/H = {1H, 2H, 4H, 7H}.
(a) Find all integers n for which (2H)n = 1H in U13/H.
(b) Find all integers n for which 2" E H.
(c) What conjecture do you have?
4. Prove: Let gH E G/H and n E Z. Then (gH)n = H if and only if g" EH.
5. Suppose [G: H] = n. Show that g" E H for all g € G.
Recall: [G: H] is the number of (left) cosets of H, which is also the size of G/H.
9.
i.e.,
#19.
Hint
inclu
10. Is th
als
11. Cc
ab
12. L
2
13.
14.
15
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