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Exercise 7.3. Let G be a finite group and n = N with n ||G|. (1) Suppose H is the unique subgroup of G with [G : H] = n. Show that H is normal in G. (2) Suppose H is a normal subgroup of G with |H| = |G|/n. Prove or disprove that H is the unique subgroup of G with [G : H] = n.

Exercise 7.3. Let G be a finite group and n = N with n ||G|. (1) Suppose H is the unique subgroup of G with [G : H] = n. Show that H is normal in G. (2) Suppose H is a normal subgroup of G with |H| = |G|/n. Prove or disprove that H is the unique subgroup of G with [G : H] = n.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 28E: 28. For an arbitrary subgroup of the group , the normalizer of in is the set . a. Prove...
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Exercise 7.3. Let G be a finite group and n = N with n ||G|. (1) Suppose H is the unique subgroup of G with [G : H] = n. Show that H is normal in G. (2) Suppose H is a normal subgroup of G with |H| = |G|/n. Prove or disprove that H is the unique subgroup of G with [G : H] = n.
Transcribed Image Text:Exercise 7.3. Let G be a finite group and n = N with n ||G|. (1) Suppose H is the unique subgroup of G with [G : H] = n. Show that H is normal in G. (2) Suppose H is a normal subgroup of G with |H| = |G|/n. Prove or disprove that H is the unique subgroup of G with [G : H] = n.
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