Chapters 1-7 Flashcards
What does |A| denote (A being a set)?
|A| denotes the size/order,
the number of elements in the set.
Define a binary operation.
If X is a set a binary operation on X is just a way of combining two elements of X into one element of X.
For example, + and × on Real Numbers are binary operations.
Formally, a binary operation on a set X is just a function from the set X × X of all pairs of elements in X to X.
Define the four Group axioms.
Define the trivial group.
The trivial group is a group that only contains the identity.
Define an abelian group
Define a power set.
Explain the notation XX.
Explain the concept of a Cayley table.
Show the Klein four Group.
Explain how Closure can be proven with a Cayley table.
Closure: There is no symbol in the body of the table that does not occur as a label.
Explain how the Identity axiom can be proven with a Cayley table.
Identity: There is an element e whose row and column are identical to the first row and column respectively.
Explain how the Inverses axiom can be proven with a Cayley table.
Inverses: e appears in every row and every column of the table; moreover, its occurrences are symmetrical with respect to the main diagonal.
Explain how the additional commutativity axiom can be proven using a Cayley table.
Abelian: The whole table is symmetrical with respect to the main diagonal.
Prove Theorem 6.2.
Let G be a group and let a,b,x ∈ G.
Right cancellativity: if ax = bx, then a = b;
Left cancellativity: if xa = xb, then a = b.
Proof:
We prove right cancellativity only: a = ae = a(xx−1) = (ax)x−1 = (bx)x−1 = b(xx−1) = be = b.
Prove Corollary 6.3.
Let G be a group. Then every element of G appears once and only once in every row and in every column of the body of the Cayley table of G.
Define Zn.
