Group: A group is a non-empty set G with binary operation satisfying:
Properties:
ab=cb abb−1=cbb−1 a(bb−1)=c(bb−1) ae=ce a=c
Uniqueness of inverse in a group:
Suppose b,c is the inverse of a in the group, since ab=e,ac=e, then ab=ac. By cancellation, b=c.
Shoe and socks property: (ab)−1=b−1a−1
Subgroup: If G is a group and H⊆G is a subset of G, then H is a subgroup of a group G if it is closed under the same binary operation as G
Cyclic subgroup: of G generated by a: Every element in G is equal to an, n>1
give an example of a group that is not cyclic: D4
Order of a Group: |G|, The number of elements of a group
Order of an Element: |g|, the smallest positive integer n such that gn=e
One-step subgroup test: Let G be a group and H a nonempty subset of G, ab−1∈H when a,b∈H, then H is a subgroup of G
Two-step subgroup test:
Center: Z(G)=a∈G|ag=ga$ for all $g∈G
Show that Z(G) is a subgroup of G
Since eg=ge=g for g∈G,
Let a1,a2∈Z(G), since a1a2g=a1ga2=g(a1a2), a1a2∈G
Let a∈Z(G), ag=ga a−1ag=a−1ga g=a−1ga ga−1=a−1g Thus, a−1∈Z(G)
Centralizer: For a in G, the centralizer of a in G, C(a) is the set of all elements in G that commute with a, C(a)=g∈G|ag=ga
Show that C(a) is a subgroup of G
Since ea=ae, e∈C(a)
Let g1,g2∈C(a), since g1g2a=g1ag2=ag1g2, g1g2∈C(a)
Let g∈C(a), and g−1 be the inverse of g in G, ag=ga g−1agg−1=g−1gag−1 g−1a=ag−1
Criterion for ai=aj: If a has infinite order, ai=aj if only if i=j. If a has finite order, then ⟨a⟩=e,…,an−1
Property of gcd: If d=gcd(a,b), there is integers t and s such as d=ta+sb
Let G be a finite cyclic group of order n, if d=gcd(k,n), then ⟨ak⟩=⟨ad⟩ and |⟨ak⟩|=nd
The fundamental theorem of cyclic groups (Theorem 4.3):
Permutation: a function from A to A that is one-to-one and onto
Permutation group: a set of permutations of A that forms a group under function composition
Symmetric group: Sn= permutations of 1,2,…,n, |Sn|=n!
Disjoint cycles commute
multiply the following two permutations in cycle notation
Write the permutation as a product of transpositions: (12)(23)…(n−1n)=(12)(13)…(1n)
Order of a permutation: The order of a permutation of a finite set is the least common multiple of the lengths of the cycles
Always even or always odd theorem for permutations: If α∈Sn, and α=t1…tk=s1…sl, then k and l are both even or odd
If An is the subset of even permutations, then |An|=|Bn|=n!2, |An| is a subgroup