Definitions:

Order: Let \(a\in G\). The \(\underline{\text{order}}\) of a, denoted ord(a) or \(|a|\), is the least positive integer n such that \(a^n=e\).

Theorems:

Theorem 3: Let G be a group and let \(a\in G\) have order \(n<\infty\). Then every power of a is equal to exactly one \(e,a,a^2,a^3,...,a^{n-1}\)

Corollary: Let ord(a)=\(n<\infty\). Then the cyclic group generated by a has n elements. \(\langle a\rangle=\{...,a^{-2},a^{-1},e,a,a^2,a^3\}=\{e,a,a^2,...,a^{n-1}\}\).

Theorem 4: Let ord(a)=\(\infty\). Then for all \(i,j\in\mathbb{Z}\), if \(i\in j\) then \(a^i\ne a^j\).

(No power of a gives the identity, and no power of a is the same.)

Corollary: If ord(a)=\(\infty\), the cyclic group generated by a has infintely many elements: \(\langle a\rangle =\{...,a^{-2},a^{-1},e,a,a^{2},...\}\)

Theorem 5: Let \(a\in G\) have order \(n<\infty\). Then \(a^m=e\) if and only if m is a multiple of n. 

Facts about cyclic groups:

Let \(G=\langle a\rangle=\{e,a,a^2,a^3,...,a^{n-1}\}\) be a cyclic group of size n. 

1. G is abelian (Since \(a^ia^j=a^{i+j}=a^{j}a^{i}\))

2. Let \(k\geq 1\) be a positive divisor of \(|G|=n\). Then G has exactly one subgroup of size k.

3. The subgroup of G of size 1 is \(\langle e\rangle =\{e\}\). If \(k>1\) is a divisor of n, the subgorup of size k is \(\langle a^{n/k}\rangle\)

Definitions

coset: Let G be a group and H a subgroup of G. For any \(a\in G\), the set \(Ha=\{ha|h\in H\}.\) This is a (right) \(\underline{\text{coset}}\) of H in G.

index: The number of cosets of H in G, n, is called the \(\underline{\text{index}}\) of H in G.

Facts about Cosets

1. All cosets have the same size as H.

2. Distinct cosets don’t overlap.

3. Every element of G is in some coset of H.

4. How should we choose a so that we don’t get repeated cosets? Choose a to be an element of G that hasn’t yet appeared in a coset.

5. How do we know when we’ve listed all cosets? We’re done when every element of G has appeared in a coset.

Theorems

Lemma: Distinct cosets don’t overlap.

Corollary: If \(a\in Hb\) then \(Ha=Hb\).

Theorem 1: The set \(\{Ha|a\in G\}\) of all cosets of H is a partiion of G.

Corollary: Define a relation \(\sim\) on G by \(a\sim b\) iff a and b are in the same coset of H.

Theorem 2: Every coset of H in G has the same size as H.

Lagrange’s Theorem: Let G be a finite group and H be a subgroup of G. Then the size of H divides the size of G.

Theorem 4: Let \(p\in\mathbb{Z}\) be prime. If \(|G|=p\), then \(G\cong\mathbb{Z}_p\).

Theorem 5: Let G be a finite group and \(a\in G\). Then ord(a) divides \(|G|\).

Definitions

Homomorphism: Let \(G_1\) and \(G_2\) be groups. A \(\underline{\text{homomorphism}}\) is a function \(f:G_1\rightarrow G_2\) such that for all \(a,b\in G_1\), \(f(ab)=f(a)f(b)\).

Homomorphic Image: If \(f:G_1\rightarrow G_2\) is an onto homomorphism, we call \(G_2\) a \(\underline{\text{homomorphic image}}\) of \(G_1\).

Kernel: Let \(f:G_1\rightarrow G_2\) be a homomorphism. The \(\underline{\text{kernel}}\) of f is the set of all elements in \(G_1\) that get sent to \(e_2\in G_2\): \(\text{ker}(f)=\{x\in G_1|f(x)=e_2\}.\)

Conjugate: Let G be a group and \(g_1x\in G\). The element \(gxg^{-1}\) is called the \(\underline{\text{conjugate}}\) of x by g. Going from x to \(gxg^{-1}\) is called \(\underline{\text{conjugation by g}}\).

Normal: Let H be a subgroup of G. We call H a \(\underline{\text{normal}}\) subgroup of G if for all \(g\in G\) and \(h\in H\), the conjugate \(ghg^{-1}\) is in H. Notation: \(H \unlhd G\)

Normal Subgroup: A subgroup H of G is called a \(\underline{\text{normal subgroup}}\) if for all \(g\in G\) and \(h\in H\). \(ghg^{-1}\in H\).

Theorems

Theorem 1: Let \(f:G_1\rightarrow G_2\) be a homomorphism. Then

1. \(f(e_1)=e_2\)

2. \(f(a^{-1})=[f(a)]^{-1}\forall a\in G_1\)

Theorem: Let \(f:G_1\rightarrow G_2\) be a homomorphism. Then f is one-to-one if and only if ker(f)=\(\{e_1\}\).

Theorem: If G is abelian, every subgroup of G is normal.

Theorem: Let H be a subgroup of G. Then H is normal in G if and only if \(aH=Ha\) for all \(a\in G\).

Theorem: A subgroup H of G is normal in G iff \(aH=Ha\forall a\in G\).

Theorem 2: Let \(f:G_1\rightarrow G_2\) be a homomorphism.

1. ran(f)=\(\{f(x)|x\in G_1\}\) is a subgroup of \(G_2\)

2. ker(f) is a normal subgroup of \(G_1\).

Definitions

coset multiplication: \(Ha\cdot Hb=H(ab)\) or \((H+a)+(H+b)=H+(a+b)\)

Theorems

Theorem 2: Let H be a normal subgroup of G. If \(Ha=Hc\) and \(Hb=Hd\), then \(H(ab)=H(cd)\).

Theorem 3: Let H be a normal subgroup of G. The set \(G/H=\{Ha|a\in G\}\)

Theorem 4: Let H be a normal subgroup of G. The function \(f:G\rightarrow G/H\) define by \[f(a)=Ha\] is a surjective homomorphsim with kernel H.

Theorem 5: Let H be a subgroup of G.

1. \(Ha=Hb\) if and only if \(ab^{-1}\in H\).

2. \(Ha=H\) if and only if \(a\in H\).

Theorems

Theorem 1: Let \(f:G\rightarrow H\) be a homomorphism with kernel K. Then \(f(a)=f(b) \Leftrightarrow Ka=Kb\).

Theorem 2: Let \(f:G\rightarrow H\) be a surjective (onto) homomorphism with kernel K. Then \(G/K\cong H\).