Quick Math Questions
Is every \( \mathbb{R} \)-linear map automatically \( \mathbb{C} \)-linear?
No, consider the complex conjugation map, which is \( \mathbb{R} \)-linear but not \( \mathbb{C} \)-linear.
Is every injective group homomorphism also surjective?
No, injectivity and surjectivity are independent. For example, the inclusion \( \mathbb{Z} \hookrightarrow \mathbb{Q} \)
is injective but not surjective (prove this).
Is every subgroup of a cyclic group also cyclic?
Yes, every subgroup of a cyclic group is cyclic. In fact, for \( \mathbb{Z}_n \), the subgroups correspond to the divisors of \( n \).
Is every integral domain a field?
No, for example \( \mathbb{Z} \) is an integral domain but not a field, since not every nonzero element has a multiplicative inverse.
If \( G \) is a group of prime order, is it necessarily cyclic?
Yes, every group of prime order is cyclic and isomorphic to \( \mathbb{Z}_p \) for some prime \( p \).
Can a ring be a field if it has zero divisors?
No, fields have no zero divisors by definition. If \( ab = 0 \) in a field, then either \( a = 0 \) or \( b = 0 \).
Can you color the natural numbers red and blue such that n and 2n are never the same color?
Yes, look at the number of 2's in prime factorization and color it red if it has an odd number and blue otherwise.
Can the function which gives you the number of isomorphism classes of groups of size n be a polynomial?
No, since that function minus 1 has infinitely many roots at every prime.