3.3 Subgroups and Cyclic Groups

Groups are the building blocks of abstract algebra. Subgroups are smaller groups within larger ones, sharing the same operation. They must include the identity element, be closed under the operation, and contain inverses for all elements.

Cyclic groups are special groups generated by a single element. Every element in a cyclic group can be expressed as a power of the generator. Cyclic groups are always abelian and can be finite or infinite in order.

Subgroups and their properties

Definition and requirements

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• A subgroup is a subset of a group that is itself a group under the same operation
• The identity element of the group must be in the subgroup ($e \in H$)
• If $a$ and $b$ are in the subgroup, then $ab$ must also be in the subgroup (closure, $a, b \in H \implies ab \in H$)
• If $a$ is in the subgroup, then $a^{-1}$ must also be in the subgroup (inverse, $a \in H \implies a^{-1} \in H$)
• The associative property holds for elements in the subgroup ($(ab)c = a(bc)$ for all $a, b, c \in H$)

Examples of subgroups

• The set of even integers under addition is a subgroup of the integers under addition ($2\mathbb{[Z](https://www.fiveableKeyTerm:z)} \leq \mathbb{Z}$)
• The set of matrices with determinant 1, known as the special linear group $SL(n, \mathbb{R})$, is a subgroup of the general linear group $GL(n, \mathbb{R})$

Subgroup identification

Steps to determine if a subset is a subgroup

• Check if the identity element of the group is in the subset ($e \in H$)
• Verify that the subset is closed under the group operation (if $a$ and $b$ are in the subset, then $ab$ must also be in the subset)
• Ensure that for each element in the subset, its inverse is also in the subset ($a \in H \implies a^{-1} \in H$)
• If all three conditions are met, the subset is a subgroup; otherwise, it is not

Examples of subgroup identification

• The set $\{1, -1, i, -i\}$ under multiplication is a subgroup of the complex numbers under multiplication
• The set of 2x2 matrices with determinant 0 is not a subgroup of $GL(2, \mathbb{R})$ because it does not contain the identity matrix

Cyclic groups and generators

Definition and properties

• A cyclic group is a group that can be generated by a single element, called a generator
• If $a$ is a generator of a cyclic group, then every element of the group can be written as a power of $a$ ($a^n$ for some integer $n$)
• The order of a cyclic group is the smallest positive integer $n$ such that $a^n = e$, where $e$ is the identity element
• A cyclic group is always abelian (commutative, $ab = ba$ for all $a, b$ in the group)
• A group of prime order is always cyclic
• A group of infinite order can be cyclic (the integers under addition, $(\mathbb{Z}, +)$)

Examples of cyclic groups

• The group of integers under addition $(\mathbb{Z}, +)$ is cyclic, with generators 1 and -1
• The group of complex nth roots of unity under multiplication is cyclic, with generator $e^{2\pi i/n}$

Identifying cyclic groups and generators

Steps to prove a group is cyclic

• To prove a group is cyclic, find an element that generates all other elements in the group
• Start by computing powers of each element in the group until all elements are generated or a repetition occurs
• If an element generates all other elements, the group is cyclic, and that element is a generator
• If no single element generates the entire group, the group is not cyclic

Finding generators in a cyclic group

• In a cyclic group of order $n$, the generators are the elements $a$ such that $gcd(|a|, n) = 1$, where $|a|$ is the order of the element $a$
• For example, in the cyclic group $\mathbb{Z}_6$ under addition, the generators are 1 and 5 because $gcd(1, 6) = 1$ and $gcd(5, 6) = 1$