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Section 10.1 Pre-read

Subsection Isomorphic groups

In many of the activities in the corresponding Discovery guide, you will be asked to verify that one group is isomorphic to another. This involves several steps:

explicitly describe a correspondence between the elements of the two groups;
convince yourself that your correspondence is bijective; and
verify that your correspondence is operation-preserving (that is, if you multiply a pair of elements in one group, and you multiply the corresponding pair of elements in the other group, the two calculation results also correspond).

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The last step might be tedious in some of the activities, so in some cases it might be appropriate to just perform some example calculations and convince yourself that all other example calculations will work similarly.

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In some activities you will be asked to verify that two groups are not isomorphic to one another. This is more subtle, as technically it would require showing that there cannot exist a bijective, operation-preserving correspondence between the two. However, isomorphic groups are essentially the “same” group in different guises, and share all the same properties. So an efficient way to demonstrate that two groups are not isomorphic is to demonstrate a property that is different between the two groups. For example,

the groups have different numbers of elements;
one group is Abelian and the other is not;
one group is cyclic and the other is not;
one group has an element of a particular order and the other group has no elements with that order;
one group has a subgroup of a particular order and the other group has no subgroups with that order;
etc..

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(Note that it is only necessary to demonstrate one property that is not shared by the two groups.)

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Subsection External products

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Definition 10.1.1. Product group.

Suppose $G$ and $H$ are groups. We can combine them into a new group $G \times H$ as the collection of all ordered pairs

(g, h),

where $g$ is a group element from $G$ and $h$ is a group element from $H .$ The group operation (in multiplicative notation) in this product group is defined by

(g_{1}, h_{1}) \cdot (g_{2}, h_{2}) = (g_{1} g_{2}, h_{1} h 2),

where in the first component of the ordered pair the group operation of $G$ is used, and in the second component the group operation of $H$ is used.

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Example 10.1.2. The plane as a product group.

Recall that if we “forget” about the operation of scalar multiplication, a vector space is always an additive group. The vector space $R^{2}$ is defined to be the collection of all ordered pairs of real numbers,

(x, y),

and addition of ordered pairs is performed component-wise:

(x_{1}, y_{1}) + (x_{2}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2}) .

In other words, as an additive group, $R^{2}$ is precisely the product group

R \times R

(hence the notation $R^{2}$ ).

Though, since this is an additive group we are talking about, it might be more appropriate to call it a sum group instead of a product group. In linear algebra, this type of “product” is also often given additive notation:

R^{2} = R \oplus R .

But in this course we will stick to multiplicative terminology and notation even when dealing with an additive group.

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Subsection Internal products

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Definition 10.1.3. Product set.

If $H$ and $K$ are two subgroups of a group $G,$ write $H K$ to mean the collection of all results of computing products

h k,

where $h$ is an element from $H$ and $k$ is an element from $K .$

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Remark 10.1.4.

In the definition of the notation $H K$ it is not really necessary that $H$ and $K$ are subgroups, just that they are both subcollections of the same group $G .$ But the subgroup case is the only one in which we will be interested.
If $G$ is an additive group, then $H K$ actually consists of all results of computing sums
$h + k$
where $h$ is from $H$ and $k$ is from $K .$ So in this context it might be more appropriate to write $H + K$ for this collection of sums instead of $H K .$

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Warning 10.1.5.

Even when $H$ and $K$ are subgroups of $G,$ the internal product set $H K$ is not necessarily a subgroup of $G!$