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2. Let G, H be Lie groups. Determine whether the following proposition is correct, and prove your conclusion. - If H is a subgroup of G, then H must be a Lie subgroup of G. - If H is a Lie subgroup of G, then the inclusion map i : H → G is an embedding if and only if i(H) is a closed subset of G. - G is a cyclic group if and only if there exists a E G such that {ak : k € Z} is dense in G.

2. Let G, H be Lie groups. Determine whether the following proposition is correct, and prove your conclusion. - If H is a subgroup of G, then H must be a Lie subgroup of G. - If H is a Lie subgroup of G, then the inclusion map i : H → G is an embedding if and only if i(H) is a closed subset of G. - G is a cyclic group if and only if there exists a E G such that {ak : k € Z} is dense in G.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 6TFE: True or false Label each of the following statements as either true or false, where is subgroup of...
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These questions are from a branch of mathematics known as Lie group theory. This is a field of study that combines algebraic structures (groups) with differential structures (manifolds). Lie groups play a crucial role in many areas of mathematics and physics, including but not limited to differential equations, quantum mechanics, and general relativity. In this theory, a "Lie algebra" is a way of describing a Lie group using vector spaces and a specific binary operation "[ , ]" to capture properties of the Lie group.
2. Let G, H be Lie groups. Determine whether the following proposition is correct, and prove your conclusion. - If H is a subgroup of G, then H must be a Lie subgroup of G. - If H is a Lie subgroup of G, then the inclusion map i : H → G is an embedding if and only if i(H) is a closed subset of G. - G is a cyclic group if and only if there exists a E G such that {ak : k € Z} is dense in G.
Transcribed Image Text:2. Let G, H be Lie groups. Determine whether the following proposition is correct, and prove your conclusion. - If H is a subgroup of G, then H must be a Lie subgroup of G. - If H is a Lie subgroup of G, then the inclusion map i : H → G is an embedding if and only if i(H) is a closed subset of G. - G is a cyclic group if and only if there exists a E G such that {ak : k € Z} is dense in G.
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