Group theory notes

Subgroup

Order of a finite group: cardinality of the set
order of an element g: least natural number of operations on g to become identity
(could be infinity)

Subgroup: a subset of the set with the same operation

1). nonempty 2). closed under operation 3). closed under inverse

trivial subgroup: identity. 
improper subgroup: itself
simple group: no proper, nontrivial (normal) subgroups

finite subgroup test: if H is a nonempty finite subset of G, and H is closed under group operation, then H is a subgroup. (skip inverse) 

Cyclic Group

if |a| is infinite, then $a^i=a^j$ only if $i=j$
if |a| is finite, then $a^i=a^j$ only if $i \equiv j $ mod |a|

collaries:
if $a^k=e$, then |a| divides k.

$<a^k>=<a^{gcd(n,k)}>$
$|a^k|=\frac{n}{gcd(n,k)}$

collaries:
|g| divides |G|. (for cyclic groups so far- Lagrange theorem.)
$<a^i>=<a^j>$, and $|a^i|=|a^j|$ iff $gcd(n,i)=gcd(n,j)$

fundamental theorem of cyclic group: every subgroup of a cyclic group is cyclic, with an order of a divisor of the order of the group. one to one correspondence between divisors and subgroups.

Subgroup lattice

The "stack" of subgroups- in a position that is a subgroup to some subgroup above

Permutation Group

order of a permutation: lcm of the order of disjoint cycles

every permutation can be written as 2-cycles (not necessarily disjoint)

even- even number of 2-cycles

Every permutation is either even or odd (Lemma: if some composition of 2-cycles is identity, then it's even)
All even permutations form a subgroup called Alternating group, consisting of exactly half of the elements of Sn (n>1)

Group Isomorphism:

There exists a function between two groups, one to one, onto, and preserve group structure ( f(g*h)=f(g)*f(h) for $g,h \in G$ )

properties: let $\psi $ be an isomorphism from G to H.
$\psi (e_{G})=e_{H}$
$\psi (g^{n})=(\psi(g))^{n}, n$ is an integer
$g, h \in G, gh=hg <=> \psi(h)\psi(g)=\psi(g)\psi(h)$
$G=<a> <=> H=<\psi(a)>$
$|a|=|\psi(a)|$
G is Abelian <=> H is Abelian
G is cyclic <=> H is cyclic
$K \in G <=> \psi(K) \in H$
The inverse of an isomorphism is an isomorphism
$K \in H, => \psi ^{-1} (K) \in G$

Every group is isomorphic to a permutation group

Group action: for example, an element in G act on every element in G to rearrange G

Coset: an element in G act on a subgroup of G. For example, gH is a left coset (of H in G containing g). 
$g\in gH$
$gH=H <=> g\in H$
$(gg')H=g(g'H)$
$gH=g'H <=> g\in g'H$
$gH=g'H$ or $gH$ and $g'H$ intersect empty
$gH=g'H <=> g^{-1}g' \in H$
$|gH|=|g'H|$
$gH=Hg <=> gHg^{-1}=H$
$gH \subseteq G <=> g \in G$

Lagrange's Theorem:
If H is a subgroup of a (finite) group G, then |H| divides |G|- the factors of |G| are the possible orders of subgroups. and $\frac{|G|}{|H|}$ is the number of left/right cosets. 

Proof: G can be partitioned by cosets of H.

[G : H] = $\frac{|G|}{|H|}$ the index of H in G: the number of left/right cosets of H in G

for all $a \in G$, |a| divides |G| since |a|=|<a>|

$a^{|g|}=e$

If H, K are subgroups of a (finite) Abelian group, then HK is also a subgroup, $|HK|=\frac{|H|*|K|}{|H\cup K|}$

G is a permutation group on a set S:
stabilizer of i: all permutations that don't move i: $stab(i)={\psi \in G : \psi(i)=i}$
orbit of i all possible positions of i: $orb(i)={\psi(i) \in S : \psi \in G}$

Orbit-stabilizer theorem: Order of a group = orb(i)*stab(i)

Normal subgroup: For all $g \in G$, gH=Hg Test: $gHg^{-1} \in H$
equivalently, (aH)(bH)=(ab)H
Every subgroup of Abelian groups are normal

Center of a group, Z(G): set of $g \in G$ such that gh=hg for all $g \in G$
Z(G) is a normal subgroup of G.

Quotient group:
Denoted as $\frac{G}{N}$, N is a normal subgroup of G. 
Set of left/right cosets of N in G, with the operation (aN)(bN)=(ab)N. ($a, b \in G$ and follows G's operation)

$\frac{Z}{nZ}$ isomorphic to Zn

If $\frac{G}{Z(G)}$ is cyclic (trivial), then G is Abelian (and countrapositive)

If [G : H] =2; then H is a normal subgroup of G.