Here are the /Solutions.

1. An unusual binary operation

For any two rational numbers a and b, define a*b = ab + a + b. Show that the rationals with this operation are a monoid but not a group.

2. Square roots

An element x of ℤ^*_m is called a square root of y mod m if x² = y (mod m). Prove that if m is odd and has at least two distinct prime factors (was: m is composite, which is not enough), then any y in ℤ^*_m either has no square roots mod m or at least four square roots mod m.

3. A big sum

Let p be prime and let a be any integer. Prove that

$\sum_{n=1}^{p-1} a^n$

is a multiple of p if and only if a ≠ 1 (mod p).

4. Bicyclic groups

Recall that a group is cyclic if every element can be written as the product of zero or more copies of some single generator g. Call a group bicyclic¹ if every element can be written as the product of some sequence of zero or more copies of two generators g and h (e.g., <>, g, h, gh, hg, g²h³g²⁷hghgh²g, etc.). Prove that S_n is bicyclic for any n > 0.

Not a real mathematical term in this context. (1)