Math 360, Fall 2016, Assignment 10

From cartan.math.umb.edu

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Carefully define the following terms, then give one example and one non-example of each:

  1. Left coset (of a subgroup HG by an element aG).
  2. Right coset.
  3. G/H (as a set).
  4. Normal subgroup.
  5. Coset multiplication.
  6. G/H (as a group).
  7. Index (of a subgroup; this is Definition 10.13 in the text).

Carefully state the following theorems (you do not need to prove them):

  1. Description of the elements of the left coset aH.
  2. Description of the elements of the right coset Ha.
  3. Theorem concerning the cardinality of the various cosets of H.
  4. Lagrange's Theorem.
  5. Theorem concerning groups of prime order.
  6. Theorem characterizing when coset multiplication is well-defined.
  7. Theorem characterizing when G/H is a group under coset multiplication.
  8. Criteria for normality.

Solve the following problems:

  1. Section 14, problems 1 and 9.
  2. Suppose that G is a group, and H is a subgroup of G with the property that, for any aG, aH=Ha. Prove that H is a normal subgroup of G.
  3. Let G be any group, and suppose that H is a subgroup of G with index two (see Definition 10.13). Prove that H is normal. (Hint: use the previous exercise. There are only two left cosets, namely eH and something else. Similarly, there are only two right cosets, namely He and something else. How are eH and He related? Therefore, how are the "something elses" related?)
  4. Consider the dihedral group Dn, and let Rn be the subgroup consisting of all rotational symmetries of the n-gon. Show that Rn is a normal subgroup of Dn. (Hint: use the previous exercise.)
  5. Find a familiar group isomorphic to the quotient group Dn/Rn. (Hint: what is the order of the quotient group?)
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