1.1 Groups and Subgroups

• A subgroup is a subset $H \subseteq G$ of a group $(G, ★)$ that satisfies closure, identity, and inverse
• What are the three conditions that a subset must satisfy to be a subgroup of a group $(G, ★)$?


  Closure, Identity, Inverse
• For a subgroup $H$, the closure property requires that for all a, b \in H</latex>, $a ★ b \in H$.
• The identity element $e$ of the group $G$ must also be in its subgroup H.
• What does the inverse condition require for a subgroup $H$?


  For each $a \in H$, $a^{ - 1} \in H$
• A subgroup $H$ of a group $G$ is also a group under the same operation $★$.
• The closure condition for a subgroup $H$ requires that a ★ b</latex> belongs to <latex>H.
• The identity element of a group must be included in every subgroup.
• Steps to verify if a subset $H$ is a subgroup of a group $(G, ★)$


  1️⃣ Check if $H$ is closed under $★$

  2️⃣ Verify that the identity element $e$ of $G$ is in $H$

  3️⃣ Ensure that for each $a \in H$, its inverse $a^{ - 1}$ is also in $H$
• What is the binary operation denoted by $★$ in Group Theory called?


  Binary operation
• The closure axiom states that for all a, b \in G</latex>, $a ★ b \in G$.
• The closure axiom states that for all $a, b \in G$, $a ★ b \in G$. For example, $2 +$$3 =$$5 \in \mathbb{Z}$ demonstrates closure under addition
• The associativity axiom requires that for all $a, b, c \in G$, $(a ★ b) ★ c =$$a ★ (b ★ c)$.
• What is the identity element in a group defined by the property $a ★ e =$$e ★ a =$$a$?


  e
• The inverse axiom states that for each $a \in G$, there exists an $a^{ - 1} \in G$ such that $a ★ a^{ - 1} =$$a^{ - 1} ★ a =$$e$. For example, $5 +$$( - 5) =$$- 5 +$$5 =$$0$ demonstrates the inverse property under addition
• A subgroup of $G$ must be a group itself under the same binary operation $★$.
• What is the identity element for the group of integers under addition ($\mathbb{Z}, +$)?


  0
• The non-zero real numbers under multiplication ($\mathbb{R} \setminus \{0\}$, $\times$) have an identity element of 1
• What is the identity element for the group of permutations of $n$ elements (S_{n}</latex>, $\circ$)?


  Identity permutation
• The closure axiom requires that for all $a, b \in G$, $a ★ b$ must be in $G$.
• What is the associativity property in Group Theory?

  (a ★ b) ★ c = a ★ (b ★ c)
• A subgroup must satisfy all four group axioms under the same binary operation as the parent group.
• The Identity axiom states that there exists an $e \in G$ such that for all $a \in G$, $a ★ e =$$e ★ a =$ a
• What is the inverse condition for a subgroup $H$?


  For each $a \in H$, $a^{ - 1} \in H$
• The set of even integers is a subgroup of the integers under addition.
• What is the trivial subgroup of the integers under addition?

  \{0\}
• The set of multiples of $n$, denoted $n\mathbb{Z}$, is a subgroup of the integers under addition.
• What is the identity element of the non-zero real numbers under multiplication?

  1
• The set of positive real numbers under multiplication forms a subgroup.
• What is the trivial subgroup of $(\mathbb{Z}, +$$)$?


  $\{0\}$
• The set of all multiples of $n$, denoted $n\mathbb{Z}$, is a subgroup because it includes $0$, is closed under addition, and has inverses for each element.
• The trivial subgroup contains only the identity element.
• Match the group with its operation and subgroups:

  $\mathbb{R} \setminus \{0\}$ ↔️ $\times$ and $\mathbb{R}^ +$

  $\mathbb{Z}$ ↔️ $+$ and $n\mathbb{Z}$
• The subgroup $\{1, - 1\}$ of $\mathbb{R} \setminus \{0\}$ is closed under multiplication and each element has an inverse within the set.
• The subgroup $\mathbb{R}^ +$ of $\mathbb{R} \setminus \{0\}$ includes $1$ and is closed under multiplication.
• What is a generating subgroup of a group $G$?


  Smallest subgroup containing $S$
• A generating subgroup must satisfy three key properties: closure, identity, and inverses.
• Steps for forming a generating subgroup:

  1️⃣ Ensure closure under products

  2️⃣ Include the identity element

  3️⃣ Verify that inverses exist
• A generating subgroup $\langle S \rangle$ includes all possible finite products of elements from S</latex> and their inverses.
• What is the generating subgroup $\langle 3 \rangle$ of $(\mathbb{Z}, +$$)$?


  $3\mathbb{Z}$