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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 Sn is bicyclic for any n > 0.

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

2014-06-17 11:57