6. Let G be a group and H a subgroup of G. Let a E G be an element Define aHa-1 = {aha-1|h E H}. A. Show that aHa-1 is also a subgroup of G. B. Suppose o(H)= n, what can you say about the order of o(aHa-1)?
6. Let G be a group and H a subgroup of G. Let a E G be an element Define aHa-1 = {aha-1|h E H}. A. Show that aHa-1 is also a subgroup of G. B. Suppose o(H)= n, what can you say about the order of o(aHa-1)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**6.** Let \( G \) be a group and \( H \) a subgroup of \( G \). Let \( a \in G \) be an element. Define \( aHa^{-1} = \{ aha^{-1} \mid h \in H \} \).
**A.** Show that \( aHa^{-1} \) is also a subgroup of \( G \).
**B.** Suppose \( o(H) = n \), what can you say about the order of \( o(aHa^{-1}) \)?
**7.** Let \( G \) be a group and \( a \in G \). The centralizer \( C(a) \) of \( a \) in \( G \) is defined as
\[
C(a) = \{ x \in G \mid xa = ax \}
\]
that is, \( C(a) \) is the set of all elements in \( G \) which commutes with \( a \). Show that \( C(a) \) is a subgroup of \( G \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba322d66-282b-4f13-8778-b2c554594cf2%2Fd7caaf32-d1fa-4704-acc1-7d91c194351d%2F1dul01q_processed.png&w=3840&q=75)
Transcribed Image Text:**6.** Let \( G \) be a group and \( H \) a subgroup of \( G \). Let \( a \in G \) be an element. Define \( aHa^{-1} = \{ aha^{-1} \mid h \in H \} \).
**A.** Show that \( aHa^{-1} \) is also a subgroup of \( G \).
**B.** Suppose \( o(H) = n \), what can you say about the order of \( o(aHa^{-1}) \)?
**7.** Let \( G \) be a group and \( a \in G \). The centralizer \( C(a) \) of \( a \) in \( G \) is defined as
\[
C(a) = \{ x \in G \mid xa = ax \}
\]
that is, \( C(a) \) is the set of all elements in \( G \) which commutes with \( a \). Show that \( C(a) \) is a subgroup of \( G \).
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