The midterm will be in class on Friday, May 2. It will cover the material in lectures up to and including Friday, April 25. That corresponds to material in the text up to and including 3.2.

Topics:

1. Know the meanings of (and be able to define) the most important concepts: group, subgroup, normal subgroup, abelian (aka commutative), homomorphism, kernel, group action, stabilizer, quotient group, coset, cyclic group, subgroup of G generated by a set S, order of a group, order of an element of a group.
2. Know how to find greatest common factors, least common multiples, the order of group element x^j (from the order of x); how to express an permutation as a product of disjoint cycles (and find its order).

Sample midterm questions:

1. Suppose f is a homomorphism from group G to group K. (a) Prove that the kernel of f is a subgroup. (b) Prove that it is a normal subgroup.
2. Consider the permution F (in S₆) that maps 1, 2, 3, 4, 5, and 6 to 4, 1, 5, 2, 6, and 3, respectively. Express F as a product of disjoint cycles, and find the order of F.
3. Let A and B be subgroups of G. (i) Prove that the intersection X of A and B is a subgroup. (ii) Prove that if A is a normal subgroup of G, then X is a normal subgroup of B.
4. List all the subgroups of the dihedral group S₃.
5. Suppose G is a group of order p², where p is prime. Prove that either G is cyclic, or else every element (except the identity) has order p.
6. Let S be a subset of the group G, and let H be the set of elements x in G such that xs=xs for every s in S. (In other words, H is the centralizer of S.) Prove that H is a subgroup of G.
7. Suppose G is a group of order 105, A is a subgroup of order 15, and B is a subgroup of order 21. Prove that the intersection must have exactly 3 elements. Prove also that every element in G can be expressed as an element of A times an element of B (i.e., as ab where a is in A and b is in B.)
8. Suppose G is a group of order 30 and that A and B are distinct subgroups each of order 3. (Distinct means that A and B are not equal.) Prove that AB is not equal to BA.

Solutions.