HINTS

Just do it.

Suppose M is such a monoid. For all x,y in M we have

use this to show that

Can a semi-group have two unit elements?

Which one of the two remains closed under its binary operation if you remove the identity?

Check associativity for all triples.

Check associativity and find an identity element.

Let M = (Z/6Z,+) and consider the submonoids M₁ = {0, 3} and M₂ = {0, 2, 4}.

Since S₁ and S₂ are closed under multiplication and since both contain the respective unit elements, their product is also closed under multiplication and contains the unit element of M₁ × M₂.

Any integral linear combination of the a_i/b_i is an integral multiple of 1/(b₁ ··· b_n).

Straightforward check.

Consider all powers of c and possible equalities between them.

Rewrite the exponent j as k + (j - k) if j k + n and replace the second term.

Check that these sets are closed under multiplication and contain the identity.

Try to make the multiplication tables.

Use the Extended Euclidean Algorithm.

Check this coordinate wise.

What if k > 0? And what if k = 0?

Does S₄ contain an element of order 12?

Use that gh = gh' implies g = g'.

Check that O_n(R) is closed under multiplication, taking inverses, and contains the identity.

Straightforward computation.

Use that disjoint cycles commute.

1. A subset of a finite set is finite.
2. There are r and s with r > s such that h^r = h^s.
3. Obvious.
4. Can you write the inverse of an element h as a power of h?

Show that the order of g is m and not smaller.

Consider the map f : Z/6Z -> Z/2Z × Z/3Z defined by f(a) = (a, a).

Does S₃ contain an element of order 6? Is S₃ commutative?

Straightforward computations.

Straightforward computations.

1. Consider the subgroup generated by (a, b). When does this subgroup contain (a, b + 1)?

   Make (1,0) and (0,1) from the two generators.

2. This comes down to solving a system of linear equations. Which one?

Check whether I is in these sets and whether they are closed under multiplication.

Consider all possible powers of 2.

Show that {0,1,2,3,4,6}, {0,1,3,5,7}, {0,1,2,4,5,6} and {0,1,2,4,6,7} are submonoids.

1. Relate to the invertible elements in the two factors.
2. Relatively prime.
3. Relatively prime.

What would be the identity? Relate associativity of *' to that of *.

1. Relatively prime to 26.
2. Save work by noticing that if a is a generator, then certain powers of it are also generators. Which powers?

1. Use the definition.
2. Use the definition.
3. There are 4 cases, determined by the image of 1.
4. Use the definition.

Suppose G is not cyclic, then all elements have order at most 2.

Just check all possible powers of an element to see whether it is a generator.

Is (2a,b) a power of (a,b)?