Problems in elementary group theory/solutions

This is the solutions page to its parent page.

True/False problems

 * 1) False: A group must contain an identity element, and is hence always nonempty.
 * 2) True: The map sends the unique element in the trivial group to the identity element of the group. The category-theoretic interpretation is that the trivial group is an initial object in the category of groups.
 * 3) True: Set-theoretically, the map sends all elements of the group to the single element of the trivial group. The homomorphism conditions are readily checked. In category
 * 4) False: Consider the group of integers (under addition) and the subset comprising nonnegative integers. This is not a subgroup, since it does not have additive inverses. However, the result is true for finite groups, and a slightly modified version is true for arbitrary groups.
 * 5) True: This boils down to proving uniqueness of identity element and inverses.
 * 6) False: The inverse map and identity element store practically no information about the group structure. In fact, the only information the inverse map carries about the group structure, is whether an element has order two.
 * 7) True:
 * 8) True:

True/False problems

 * 1) True: This follows by cardinality counting, and the fact that any two subgroups must intersect at at least one element. The statement is also true for infinite groups, by reduction to the finite case.
 * 2) False: The elementary Abelian group of order $$9$$ is a union of four proper subgroups of order three.
 * 3) False: The reason why we can't prove it is clear (once one tries to prove it) though coming up with counterexamples requires a better understanding of group structures.
 * 4) False: The result would be true if we assumed $$\varphi$$ to be surjective.
 * 5) False: Again, it is clear that one cannot prove this. A counterexample is the symmetric group of order three, which is a product of its Sylow 2-subgroup and its Sylow 3-subgroup.
 * 6) True: This uses the fact that the inverse map is involutive.
 * 7) False: It is true, though, that the union of all conjuates of a proper subgroup can never be the whole group.