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A group G is called metacommutative if there exists an exact sequence 1 → G₁ →G→ G₂ → 1 where G₁ and G₂ are commutative (see the first hand in for the notion of an exact sequence). (a) Prove that a subgroup of a metacommutative group is metacommutative. (b) Show that there exist metacommutative non-commutative groups of order 2n for all n. Do there exist metacommutative, non-commutative groups of any finite order? (c) Suppose that H is the smallest subgroup of G containing all elements of the form aba¯¹6-¹ with a, b € G. Prove that G is metacommutative if and only if H is commutative.

A group G is called metacommutative if there exists an exact sequence 1 → G₁ →G→ G₂ → 1 where G₁ and G₂ are commutative (see the first hand in for the notion of an exact sequence). (a) Prove that a subgroup of a metacommutative group is metacommutative. (b) Show that there exist metacommutative non-commutative groups of order 2n for all n. Do there exist metacommutative, non-commutative groups of any finite order? (c) Suppose that H is the smallest subgroup of G containing all elements of the form aba¯¹6-¹ with a, b € G. Prove that G is metacommutative if and only if H is commutative.

Advanced Engineering Mathematics
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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G is called metacommutative if there exists an exact sequence 1 → G₁ →G → G₂ → 1 where G₁ and G2 are commutative (see the first hand in for the notion of an exact sequence). (a) Prove that a subgroup of a metacommutative group is metacommutative. (b) Show that there exist metacommutative non-commutative groups of order 2n for all n. Do there exist metacommutative, non-commutative groups of any finite order? A group (c) Suppose that H is the smallest subgroup of G containing all elements of the form aba-¹6-¹ with a, b E G. Prove that G is metacommutative if and only if H is commutative. 1
Transcribed Image Text:G is called metacommutative if there exists an exact sequence 1 → G₁ →G → G₂ → 1 where G₁ and G2 are commutative (see the first hand in for the notion of an exact sequence). (a) Prove that a subgroup of a metacommutative group is metacommutative. (b) Show that there exist metacommutative non-commutative groups of order 2n for all n. Do there exist metacommutative, non-commutative groups of any finite order? A group (c) Suppose that H is the smallest subgroup of G containing all elements of the form aba-¹6-¹ with a, b E G. Prove that G is metacommutative if and only if H is commutative. 1
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