think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal and give a counterexample to show that a normal subgroup need not be characteristic.
think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal and give a counterexample to show that a normal subgroup need not be characteristic.
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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![4. A subgroup is called characteristic if it is invariant under all automorphisms.
We write H char G if 0(h) € H for all h H and 0 € Aut(G). We may
think of this as being a stronger type of normality. Prove that a characteristic
subgroup is normal and give a counterexample to show that a normal subgroup
need not be characteristic.
Hint: In an abelian group all subgroups are normal. Are they call characteris-
tic?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcba284a9-69bd-43db-90ac-9d691f0cd71a%2Fdd193641-3a6f-4cef-b72f-0bbfec2d514a%2Fp70kywg_processed.png&w=3840&q=75)
Transcribed Image Text:4. A subgroup is called characteristic if it is invariant under all automorphisms.
We write H char G if 0(h) € H for all h H and 0 € Aut(G). We may
think of this as being a stronger type of normality. Prove that a characteristic
subgroup is normal and give a counterexample to show that a normal subgroup
need not be characteristic.
Hint: In an abelian group all subgroups are normal. Are they call characteris-
tic?
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