Question

Give and example of a group and n such that the set of elements whos nth power is e do not form a subgroup

Answer 1

An example in group theory where the set of elements whose nth power is the identity does not form a subgroup is a group with elements {e, a, b, c}, where a and b squared equal e, but their product does not square to e, violating the closure property.

Step-by-step explanation:

In the context of group theory in mathematics, a subgroup is a smaller group consisting of elements of a larger group that satisfy the group axioms. To provide an example where the set of elements whose nth power is the group's identity element e does not form a subgroup, let's consider the group G = {e, a, b, c} under some binary operation ∗ and let's assume that when we compute the nth power of these elements where n = 2, we find:

• a₂ = e
• b₂ = e
• c₂ ≠ e

Also, assume that when we combine a and b we get c, that is, a ∗ b = c. Now, the elements whose squares (2nd power) are e are a and b. However, since a ∗ b = c, and c₂ is not e, this set {a, b} does not satisfy the closure property and thus can't be a subgroup.

Answer 2

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