- Closed
- Associative
- Identity
- Inverse
- A subset H is closed under the binary operation of G
- H with the induced operation of G is itself a group
A subset H of a group G is a subgroup of G <=>
- H is closed under the binary operation of G
- the identity element of G is in H
- for all a in H, the inverse of a is also in H
H = { a^n : n in Z } is the smallest subgroup containing a
- every subgroup containing a contains H
- H is cyclic
- denoted by
- A group G is cyclic if G = = { a^n : n in Z } for some a in G
- a is a generator of G
- if has finite number of element, order of a = ||
- if has infinite number of element, order of a is of infinite
- Every cyclic group is abelian
- A subgroup of a cyclic group is cyclic
- A cyclic group is isomorphic to <Z, +> if it is of infinite order
- A cyclic group is isomorphic to <Zn, +n> if it is of finite order
- if there exists an isomorphism between two groups, then the isomorphism is also shared among their subgroups.
- i.e. to understand the structures of cyclic groups, it suffices to study those of Z and Zn.
Let G be a cyclic group of order n generated by a. Let b in G, b = a^s, d = gcd(n, s)
= { e, a, a^d, a^2d, ... , a^n-d }
<a^r> = <a^s> iff gcd(r, n) = gcd(s, n)
If a is a generator of a finite cyclic group of order n, then other generators are of the form a^r, where r is relatively prime ti n.
The number of subgroups of a cyclic group of order n is equal to the number of divisors of n.
Let A be a set, a permutation is a 1-1 and onto function from A to A
Let A be a nonempty set. Then the set Sa of all permutations of A is a group under function composition.
Sn is nonabelian for all n >= 3.
Every group is isomorphic to a subgroup of a group of permutations.
Let p be a permutation, the relation ~ defined by a ~ b <=> a = p^n(b) for some integer n is an equivalence relation
The equivalence classes determined by the above equivalence relation are the orbits of p.
- A permutation p is a cycle if it has at most one orbit containing more than one element.
- The length of a cycle is the number of elements in the largest orbit.
Every permutation p of a finite set is a product if disjoint cycles.
The multiplication if disjoint cycles are commutative.
A transposition is a cycle of length 2.
Any permutation of a finite set of at least two elements is a product of transposition.
No permutation in Sn can be expressed both as a product of an even number of transpositions and as a product of an odd number of transpositions.
If n >= 2, then the set An of all even permutations of {1..n} forms a subgroup of order n!/2 of Sn
The subgroup of Sn consisting of the even permutations of n letters is the alternating group An on n letters.
Let H < G, let ~L be a ~L b <=> a'b in H ~L and ~R are both equivalence relations on G
- Left coset: aH = { ah : h in H }
- Right coset: Ha = { ha : h in H }
- If a group G is abelian then each left coset is also right coset.
- If a group G is abelian then each left coset may or may not right coset.
Let H be a subgroup of a finite group G. Then the order of H divides the order of G.
Let H be a subgroup of a finite group G. Then every coset has the same number of elements as H.
The order of an element of a finite group divides the order of the group.
Every group of prime order is cyclic.
Let H be a subgroup of a group G. The number of left cosets of H is the index of H in G, and is denoted by [G:H]
[G:H] = [G]/[H]
Suppose that H and K are subgroups of a group G such that K <= H <= G and [G:H] and [H:K] are finite then [G:K] is also finite and [G:K] = [G:H][H:K]
fuck you
The group Zm x Zn is cyclic and isomorphic to Zmn <=> m and n are relatively prime
The group of sum of Zmi is cyclic and isomorphic to Zmmmmm <=> the numbers m are pairwise coprime
order of an element of a direct product of groups = lcm ...