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We say a group G is a cyclic group if it can be generated by a single element.
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To understand this definition and notation, we must first explain what it means
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for a group to be generated by an element.
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Once we’ve done that, we’ll give several examples,
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explain why the word “cyclic” was chosen for this definition,
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and then finally talk about why these types of groups are so important.
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When working with groups, you typically use additive notation or multiplicative notation.
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This is done even if the elements of the group are not numbers and the group operation
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is not numerical, but is instead something like geometric transformations or function composition.
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When using additive notation, the identity element is denoted by 0,
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and when using multiplicative notation, the identity element is denoted by 1.
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But keep thinking abstractly,
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even if the notation tries to lure your mind into the familiar realm of the real numbers…
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Let’s now dive into the definition of cyclic groups.
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Let G be any group, and pick an element 'x' in G.
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Here’s a puzzle: what’s the smallest subgroup of G that contains 'x'?
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First, any subgroup that contains 'x' must also contain its inverse…
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It also has to contain the identity element…
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And to be closed under the group operation, it has to contain all powers of 'x'...
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and all powers of the inverse of 'x'...
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This set of all integral powers of 'x' is the smallest subgroup of G containing 'x'.
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We call it the group generated by 'x' and denote it using brackets.
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If G contains an element 'x' such that G equals the group generated by 'x',
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then we say G is a cyclic group.
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It’s worth taking a moment to repeat this definition using additive notation.
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Let H be a group, and pick an element 'y' in H.
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The group generated by 'y' is the smallest subgroup of H containing 'y'.
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It must contain 'y', its inverse '-y', and the identity element 0.
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And to be a group it must contain all positive and negative multiples of 'y'.
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If H can be generated by an element 'y', then we say H is a cyclic group.
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Let’s look at a few examples of cyclic groups.
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A classic example is the group of integers under addition.
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The integers are generated by the number 1.
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To see this, remember the group generated by 1 must contain:
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1, the identity element 0, the additive inverse of 1 (which is -1),
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and it must also contain all multiples of 1 and -1.
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This covers all the integers.
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The integers are a cyclic group!
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The integers are an example of an infinite cyclic group.
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Let’s now look at a FINITE cyclic group.
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The classic example is the integers mod N under addition.
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This is a finite group with N elements.
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It is also generated by the number 1.
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But something different happens here.
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Look at all the positive and negative multiples of 1.
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Recall that 'n' is congruent to 0 mod 'n'…
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n + 1 is congruent to 1 Mod 'n', and so on.
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-1 is congruent to N-1, -2 is congruent to N-2, and so on..
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So the group generated by 1 repeats itself.
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It cycles through the numbers 0 through N-1 over and over.
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This is why it’s called a “cyclic group.”
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The integers mod N are a finite, cyclic group under addition.
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In abstract algebra, the integers mod N are written like this.
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This will make sense once you’ve studied quotient groups,
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so don’t panic if you're not familiar with this notation.
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We’ve now seen two types of cyclic groups: the integers Z under addition, which is infinite,
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and the integers mod N under addition, which is finite.
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Are there other cyclic groups?
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No! This is it!
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The complete collection of cyclic groups.
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The integers.
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The integers mod 2.
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The integers mod 3…
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The integers mod 4, and so forth.
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Oh, and don’t forget the trivial group.
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Why are cyclic groups so important?
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The big reason is due to a result known as
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The Fundamental Theorem of Finitely Generated Abelian Groups
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That’s quite a title!
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What it says is that any abelian group that is finitely generated can be broken apart
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into a finite number of cyclic groups.
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And every cyclic group is either the integers, or the integers mod N.
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So cyclic groups are the fundamental building blocks for finitely generated abelian groups.
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It takes a lot of work to understand and prove this theorem,
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but you’ve just taken your first step…
