Question

Let \(G=\langle a\rangle\) be a cyclic group of order 60 , and \(H=\left\langle a^{35}\right\rangle\). List all the left cosets of \(H\) in \(G\).

Short answer

H has 12 distinct left cosets in G: \(a^0H, a^1H, \ldots, a^{11}H\).

Step-by-step solution

Determine the Order of the Subgroup

The order of a subgroup generated by an element in a cyclic group is found by taking the order of the group and dividing it by the greatest common divisor (GCD) of the order of the group and the exponent of the element generating the subgroup. First, we must find this order. \[ |H| = \frac{|G|}{\gcd(n, 35)} = \frac{60}{\gcd(60, 35)} \] By finding the GCD of 60 and 35: \[ \gcd(60, 35) = 5 \] Therefore, the order of the subgroup \(H\) is: \[ |H| = \frac{60}{5} = 12 \]

List the Elements of Subgroup H

The subgroup \(H\) is generated by \(a^{35}\). Since \(|H| = 12\), the elements of \(H\) will be \(a^{35k}\) for \(k = 0, 1, 2, \ldots, 11\). Therefore, \(H\) consists of the following elements: \[ H = \{ e , a^{35}, a^{70}, a^{105}, a^{140}, a^{175}, a^{210}, a^{245}, a^{280}, a^{315}, a^{350}, a^{385} \} \] Each of these elements is congruent to a power of \(a\): Since \(60\equiv 0\mod 60\), \(a^{60} = e\). So, \(a^{k}\) for \(k \geq 60\) reduces modulo 60.

Identify Distinct Left Cosets

A left coset of \(H\) in \(G\), denoted \(gH\), is given by \(gH = \{gh \mid h \in H\}\). Since \(G = \langle a \rangle\), cosets are of the form \(a^kH\). To find all distinct cosets, we must use powers of \(a\) such that \(a^k \) is not previously covered by cosets due to subgroup periodicity. The representative elements are: \[ a^0, a^1, a^2, \ldots, a^{11} \] because \(|H|=12\).

Construct and List Each Coset

Each distinct left coset \(a^kH\) is formed by multiplying every element of \(H\) by \(a^k\). The distinct left cosets are: 1. \(a^0H = H = \{e, a^{35}, a^{10}, a^{45}, \ldots \}\) 2. \(a^1H = \{a^1, a^{36}, a^{11}, \ldots \} \) 3. \(a^2H = \{a^2, a^{37}, a^{12}, \ldots \} \) Continue pattern... 11. \(a^{11}H = \{a^{11}, a^{46}, a^{21}, \ldots \} \)These form all 12 left cosets because 60/5 = 12.

Key concepts

Group Theory

Group theory is a mathematical framework that studies structures known as groups. A group consists of a set, combined with an operation that satisfies four key properties: closure, associativity, identity, and invertibility. These properties mean:

• Closure: If two elements are in the group, the operation on these elements will also result in an element in the group.
• Associativity: The result does not change irrespective of how the operation is grouped; meaning \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity: There exists an element in the group that does not change other elements when the operation is applied.
• Invertibility: For every element in the group, there is another element that combines to form the identity element.

In the context of the exercise, we focus on cyclic groups which are special types of groups generated by repeatedly applying the group operation to a single element. This structure greatly simplifies calculations, as we only consider the powers of the generator element.

Cosets

Cosets are fundamental in understanding group partitioning. Within a group, they can be regarded as "shifts" over a subgroup. Given a subgroup \(H\) and an element \(g\) in the group \(G\), the left coset \(gH\) consists of all elements formed by multiplying \(g\) by each member of \(H\). More formally, \[ gH = \{gh \mid h \in H\} \]
In our exercise, we form cosets by taking each representative from the cyclic group \(G\) and applying it across the subgroup \(H\). This approach ensures we cover the group in a structured manner without repetition. The distinct cosets collectively represent a partition of \(G\), explaining the group's structure in smaller, manageable parts tied to the subgroup \(H\).

Subgroups

A subgroup is a subset of a group that itself forms a group with the operation inherited from the larger group. In our exercise, \(H\) is the subgroup of the cyclic group \(G\). Subgroups are important because they help break down complex group structures into smaller, more comprehensible units. This is crucial when working with large groups since it allows using the properties of the subgroup to make calculations more manageable.

To determine a subgroup like \(H\), in a cyclic group, it is often generated by a power of the group's generator. The order or size of the subgroup is particularly significant and can be computed using the order of the group and the greatest common divisor (GCD) of the group order and the subgroup's generator exponent. Such calculations are fundamental for operations in group theory like listing cosets as illustrated in the exercise.

Greatest Common Divisor

The greatest common divisor (GCD) is a basic yet significant concept in number theory, measuring the largest integer that can divide two numbers without leaving a remainder. When applied to group theory, particularly cyclic groups, it's invaluable for determining subgroup orders.

In our context, you calculate the subgroup's order in a cyclic group by dividing the total order of the group by the GCD of the group order and the exponent related to the subgroup generator. For this reason, understanding and computing GCD efficiently allows for defining subgroup elements and cosets, as detailed in the exercise.

• Example: The GCD of \(60\) and \(35\) is calculated to be \(5\). This means the power of \(a^{35}\), or equivalently the cyclic nature of the group, 'resets' every 5 steps within the group \(G\).

Learning to find the GCD is crucial not only in group theory but also in many areas like simplifying fractions and computing least common multiples (LCM). In group contexts, it is especially pivotal for understanding the "size" and "shape" of subgroups.