Group Theory Calculator

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Example

Created on 2024-06-20Asked by Liam Lopez (Solvelet student)

Prove that the group

(\mathbb{Z}_n, +)

is cyclic.

Solution

To prove that the group

(\mathbb{Z}_n, +)

is cyclic: A group

(\mathbb{Z}_n, +)

is cyclic if there exists an element

g \in \mathbb{Z}_n

such that every element of

\mathbb{Z}_n

can be written as

g^k

for some integer

k

. Consider the element

1 \in \mathbb{Z}_n

. We need to show that every element of

\mathbb{Z}_n

can be written as a multiple of 1:

\{0, 1, 2, \ldots, n-1\} = \{0 \cdot 1, 1 \cdot 1, 2 \cdot 1, \ldots, (n-1) \cdot 1\}

Clearly, multiplying 1 by any integer

k

and reducing modulo

n

gives every element of

\mathbb{Z}_n

1 \cdot k \mod n = k \mod n

Therefore, the element 1 generates the entire group

\mathbb{Z}_n

, making it cyclic. Hence,

(\mathbb{Z}_n, +)

is a cyclic group. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Elizabeth Thomas on Solvelet

1. Prove that every subgroup of a cyclic group is cyclic,2. Show that the set of invertible

2\times2

matrices with real entries forms a group under matrix multiplication.,

DefinitionGroup theory is the study of algebraic structures in abstract algebra called groups. A group is a set with a binary operation which combines any two elements to form a third element. Every group must be subject to four conditions called “group axioms”, namely closure, associativity, identity and invertibility. A group theory is a method to represent algebraic terms and the perumutation of base structures, and symmetries and added structures in the study of various fields like physics, chemistry and mathematics. The group theory is of fundamental important in the study of abstract algebra and number theory. Example: The set of integers Z with addition is a group, it has a identity 0 and all of its elements have inverses, ie. the negative of all elements.

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