where is an ordering of the elements being permutated by the elements of , is the permutation ‘shifting’ all these elements to their successor, and is the permutation swapping the first 2 of these.

I won’t provide a formal proof here as it’s a little tedious, but you can easily convince yourself that every permutation of these elements is a combination of these 2 permutations, because alternating between them in succession allows you to move any of the elements to an arbitrary position in the ordering.

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.

That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.

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In particular, if is a group isomorphism, then

- is abelian is abelian
- for all .

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are bijective.

Consider the function . By the left-cancellation property of groups, if , then . This means that if , then , and it follows that is injective. Since the domain and codomain of are the same (), this means that is in fact bijective.

The proof for bijectivity of is identical, but uses the right-cancellation property instead.

The intuitive significance of this is that left- or right-multiplication by a given element always sends distinct elements to distinct elements, which is pretty neat.

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- if and , then
- if , then there is a such that (meaning has no maximum element)

A cut can be equivalently determined solely by alone, rather than the pair .

Cuts can then be used to construct by defining any to be the cut where is the set of all members of such that . That is, is simply defined as the subset of rationals smaller than itself, and is the set of all such subsets.

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- boolean
- byte
- short
- char
- int
- long
- float
- double

All other types not in this list are reference types.

A value with a primitive type stores the information determining the value itself, such as the binary code for a given integer or character. By contrast, a value with a reference type stores the *address* of the location in memory where this information can be found. For instance, after declaring and instantiating a variable

the variable doesn’t store the code for the String itself, but rather a ‘pointer’ to where this code can be found.

This has important consequences for how primitive values are assigned to variables or passed as parameters, as opposed to reference values. A primitive value passed or assigned will have the bits describing the value itself copied to the target, whereas a reference value will merely have the *address* of these bits copied.

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- a surface is non-orientable if and only if it has the Möbius strip as a topological subspace, and
- the Möbius strip is the
*only*surface with this property.

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