Problems in elementary group theory/solutions

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This is the solutions page to its parent page.

Definition understanding

True/False problems

  1. False: A group must contain an identity element, and is hence always nonempty. Relevant terms: group
  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. Relevant terms: trivial group, homomorphism 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 categoryRelevant terms: homomorphism of groups, trivial group
  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. Relevant terms: subgroup Relevant facts: sufficiency of subgroup condition
  5. True: This boils down to proving uniqueness of identity element and inverses. Relevant facts: equivalence of definitions of group
  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.

Fact understanding

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. Relevant facts: Union of two subgroups is not a subgroup, Union of all conjugates is proper
  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. Relevant terms: normal subgroup, 2-subnormal subgroup, subnormal subgroup, transitive subgroup property Relevant facts: normality is not transitive, characteristic of normal implies normal
  4. False: The result would be true if we assumed \varphi to be surjective. Relevant terms: normal subgroup Relevant facts: normality is image-closed
  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. Relevant terms: permutably complemented subgroup, product of subgroups
  6. True: This uses the fact that the inverse map is involutive. Relevant terms: permuting subgroups, product of subgroups
  7. False: It is true, though, that the union of all conjuates of a proper subgroup can never be the whole group. Relevant terms: conjugate subgroups, normal subgroup Relevant facts: union of all conjugates is proper