(1) Compute the fundamental group of X = R³\{x – axis}U {y – axis} U{z – axis}.
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Algebraic topology
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- 13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.26. Prove or disprove that if a group has an abelian quotient group , then must be abelian.
- Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.
- Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.