⭕Groups and Geometries Unit 3 – Cosets and Lagrange's Theorem

Key Concepts

• Cosets partition a group into disjoint subsets of equal size
• Lagrange's Theorem relates the order of a group to the order of its subgroups
• Left cosets and right cosets are defined using the operation of the group
  • Left coset: $aH = \{ah : h \in H\}$ where $a \in G$ and $H \leq G$
  • Right coset: $Ha = \{ha : h \in H\}$ where $a \in G$ and $H \leq G$
• The index of a subgroup $H$ in a group $G$, denoted $[G:H]$, is the number of distinct left (or right) cosets of $H$ in $G$
• Lagrange's Theorem states that for a finite group $G$ and a subgroup $H$, the order of $H$ divides the order of $G$
  • Mathematically: $|G| = [G:H] \cdot |H|$
• Cosets can be used to prove certain subgroups are normal subgroups
• The order of an element $a$ in a group $G$, denoted $|a|$, divides the order of the group $|G|$

Group Fundamentals

• A group $(G, *)$ is a set $G$ together with a binary operation $*$ that satisfies four axioms:
  • Closure: For all $a, b \in G$, $a * b \in G$
  • Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
  • Identity: There exists an element $e \in G$ such that for all $a \in G$, $a * e = e * a = a$
  • Inverses: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$
• The order of a group $G$, denoted $|G|$, is the number of elements in the group
• A group is called abelian (or commutative) if for all $a, b \in G$, $a * b = b * a$
• Examples of groups include:
  • $(\mathbb{Z}, +)$: the integers under addition
  • $(\mathbb{R} \setminus \{0\}, \cdot)$: the non-zero real numbers under multiplication
  • $(\mathbb{Z}_n, +_n)$: the integers modulo $n$ under addition modulo $n$
• The identity element and inverse elements are unique in a group

Subgroups Explained

• A subgroup $H$ of a group $G$ is a non-empty subset of $G$ that forms a group under the same operation as $G$
• For a subset $H$ of a group $G$ to be a subgroup, it must satisfy the following conditions:
  • Closure: For all $a, b \in H$, $a * b \in H$
  • Identity: The identity element $e$ of $G$ is in $H$
  • Inverses: For each $a \in H$, $a^{-1} \in H$
• Every group $G$ has at least two subgroups: the trivial subgroup $\{e\}$ and the group $G$ itself
• The center of a group $G$, denoted $Z(G)$, is the set of elements that commute with every element in $G$
  • $Z(G) = \{a \in G : ax = xa \text{ for all } x \in G\}$
  • The center is always a subgroup of $G$
• A subgroup $N$ of a group $G$ is called a normal subgroup if for all $g \in G$ and $n \in N$, $gng^{-1} \in N$
  • Normal subgroups are important for constructing quotient groups

Cosets Defined

• Let $G$ be a group and $H$ a subgroup of $G$. For an element $a \in G$, the left coset of $H$ with representative $a$ is the set:
  • $aH = \{ah : h \in H\}$
• Similarly, the right coset of $H$ with representative $a$ is the set:
  • $Ha = \{ha : h \in H\}$
• The collection of all left cosets of $H$ in $G$ forms a partition of $G$
  • Every element of $G$ belongs to exactly one left coset of $H$
  • The same holds true for right cosets
• If $G$ is an abelian group, then left cosets and right cosets coincide
• The number of distinct left (or right) cosets of $H$ in $G$ is called the index of $H$ in $G$, denoted $[G:H]$
• Examples:
  • In $(\mathbb{Z}, +)$, the subgroup $2\mathbb{Z}$ (even integers) has two cosets: $2\mathbb{Z}$ (even integers) and $1 + 2\mathbb{Z}$ (odd integers)
  • In the dihedral group $D_4$ (symmetries of a square), the subgroup $R$ (rotations) has two cosets: $R$ and $sR$ (where $s$ is any reflection)

Properties of Cosets

• Left cosets (or right cosets) of a subgroup $H$ in a group $G$ are either identical or disjoint
  • If $aH \cap bH \neq \emptyset$, then $aH = bH$
• The left coset $aH$ is equal to the right coset $Ha$ for all $a \in G$ if and only if $H$ is a normal subgroup of $G$
• For any $a, b \in G$ and subgroup $H$ of $G$:
  • $aH = bH$ if and only if $a^{-1}b \in H$
  • $Ha = Hb$ if and only if $ab^{-1} \in H$
• The product of two cosets $aH$ and $bH$ is defined as:
  • $aH \cdot bH = \{ah \cdot bk : h, k \in H\}$
  • If $H$ is a normal subgroup of $G$, then $aH \cdot bH = abH$
• The set of all left cosets of $H$ in $G$ forms a group under the operation $(aH)(bH) = abH$ if and only if $H$ is a normal subgroup of $G$
  • This group is called the quotient group (or factor group) of $G$ by $H$, denoted $G/H$

Lagrange's Theorem

• Lagrange's Theorem states that for a finite group $G$ and a subgroup $H$ of $G$, the order of $H$ divides the order of $G$
  • Mathematically: $|G| = [G:H] \cdot |H|$
• Consequences of Lagrange's Theorem:
  • The order of any element $a$ in a finite group $G$ divides the order of $G$
  • If $G$ is a finite group and $a \in G$, then $a^{|G|} = e$
  • If $G$ is a finite group of prime order, then $G$ is cyclic and has no proper subgroups
• Lagrange's Theorem can be used to prove that certain groups are not subgroups of others
  • Example: $\mathbb{Z}_4$ cannot be a subgroup of $\mathbb{Z}_6$ because $4$ does not divide $6$
• The converse of Lagrange's Theorem is not true in general
  • Example: $A_4$ (the alternating group on 4 elements) has order $12$, but no subgroup of order $6$

Applications and Examples

• Cosets and Lagrange's Theorem have numerous applications in group theory and related fields:
  • Proving certain groups are not subgroups of others (using Lagrange's Theorem)
  • Classifying all subgroups of a given finite group (using Lagrange's Theorem to limit possibilities)
  • Constructing quotient groups (using cosets of normal subgroups)
  • Proving Cauchy's Theorem: If $G$ is a finite group and $p$ is a prime dividing $|G|$, then $G$ has an element of order $p$
  • Proving Sylow's Theorems: Existence and properties of subgroups of order $p^k$ (where $p$ is prime) in finite groups
• Example: In the dihedral group $D_4$ (symmetries of a square), Lagrange's Theorem can be used to:
  • Prove that $D_4$ has no subgroup of order $3$ (since $3$ does not divide $8$, the order of $D_4$)
  • Classify all subgroups of $D_4$ (limiting possibilities to orders $1$, $2$, $4$, and $8$)
• Example: In the group of integers modulo $12$ under addition ($\mathbb{Z}_{12}$), cosets can be used to:
  • Partition $\mathbb{Z}_{12}$ into cosets of the subgroup $\{0, 4, 8\}$: $\{0, 4, 8\}$, $\{1, 5, 9\}$, $\{2, 6, 10\}$, and $\{3, 7, 11\}$
  • Construct the quotient group $\mathbb{Z}_{12} / \{0, 4, 8\}$, which is isomorphic to $\mathbb{Z}_4$

Common Pitfalls

• Confusing left cosets and right cosets
  • Remember: $aH = \{ah : h \in H\}$ (left coset) and $Ha = \{ha : h \in H\}$ (right coset)
• Assuming that left cosets and right cosets are always equal
  • They are equal for all $a \in G$ if and only if $H$ is a normal subgroup of $G$
• Forgetting that the index $[G:H]$ is the number of distinct cosets, not the order of a coset
  • Each coset has the same order as the subgroup $H$
• Misapplying Lagrange's Theorem
  • The converse is not true: if $d$ divides $|G|$, there may not be a subgroup of order $d$
  • Example: $A_4$ has order $12$ but no subgroup of order $6$
• Assuming that all subgroups of a group are normal
  • Example: In $S_3$, the subgroup $\{(1), (1 2)\}$ is not normal
• Attempting to construct a quotient group using cosets of a non-normal subgroup
  • The cosets must be of a normal subgroup for the quotient group to be well-defined
• Confusing the order of an element $a$ (smallest positive integer $n$ such that $a^n = e$) with the order of the subgroup generated by $a$ (which is equal to the order of $a$)