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Section 7.2 Discovery guide

Subsection Examples

Discovery 7.1.

Recall that \(S_n\) denotes the group of permutations of the set of numbers

\begin{equation*} \{1,2,\dotsc,n\} \text{.} \end{equation*}

In Chapter 6 of the textbook the author introduced cycle notation for cycles in a permutation group. For example, \(\threecycle{1}{2}{3}\) denotes the \(3\)-cycle

\begin{equation*} 1 \mapsto 2 \mapsto 3 \mapsto 1 \text{.} \end{equation*}
(a)

In \(S_3\text{,}\) write \(r\) to represent the \(3\)-cycle \(\threecycle{1}{2}{3}\) and \(s\) to represent the \(2\)-cycle \(\twocycle{1}{2}\text{.}\) And, as usual, write \(e\) to represent the identity element (the trivial permutation).

Using only formulas in the symbols \(e, r, s\text{,}\) fill in the multiplication table for \(S_3\text{.}\) Remember that products should be computed in the order left-hand-reference-column-entry-times-top-reference-row-entry, but that in a product of permutations the factor permutations are applied/analyzed right-to-left.

\(\times\) e r ? s ? ?
e
r
?
s
?
?
Figure 7.2.1.
(c)

Propose an Isomorphism between \(S_3\) and \(D_3\text{.}\) Describe your isomorphism without using the letters \(r\) or \(s\text{.}\) (Remember that to describe a function, you need to describe how inputs are turned into outputs. This doesn't have to be a formula like in calculus class, it can simply be a list of input-output pairs.)

Discovery 7.2.

Propose an Isomorphism between the additive group \(\R\) and the multiplicative group of matrices explored in Discovery 2.5. Which task in Discovery 2.5 addresses the operation-preserving part of the definition of isomorphism?

Discovery 7.3.

Propose an isomorphism between the additive group \(\R\) and the multiplicative group \(\R^{\mathrm{pos}}\text{.}\) What is the inverse mapping of your isomorphism?

Hint.

Do you know a special function \(f(x)\) from first-year calculus that satisfies

\begin{equation*} f(a+b) = f(a) f(b)\text{?} \end{equation*}

Discovery 7.4.

(a)

Convince yourself that every group is isomorphic to itself.

Hint.

By definition of Isomorphic groups, for an arbitrary group \(G\) you must come up with a way to describe an isomorphism \(\funcdef{\varphi}{G}{G}\text{.}\)

(b)

Come up with two different examples of isomorphisms \(\Z \to \Z\text{.}\)

Subsection Some properties preserved by isomorphisms

For discovery activities in this subsection, assume \(G\) and \(G'\) are groups and that \(\funcdef{\varphi}{G}{G'}\) is an isomorphism. In case you haven't taken AUMAT 250, the notation \(\funcdef{\varphi}{G}{G'}\) means that \(\varphi\) is the function, \(G\) is the collection of “inputs,” and \(G'\) is the collection of “potential outputs,” so that for input element \(x\) in \(G\text{,}\) the corresponding output \(\varphi(x)\) is an element of \(G'\text{.}\)

We will treat both \(G\) and \(G'\) as multiplicative groups, but we will not introduce any special symbols to distinguish between the two group operations — you should use context to help you decide if an expression involves multiplication in \(G\) or multiplication in \(G'\text{.}\) But we will distinguish between the two different identity elements: write \(e\) for the identity element in \(G\) and \(e'\) for the identity element in \(G'\text{.}\)

For reference, here are the relevant definitions.

Discovery 7.5. Isomorphism preserves identity.

(a)

Argue that for all group elements \(y\) in \(G'\text{,}\) we have

\begin{equation*} \varphi(e) \cdot y = y \text{.} \end{equation*}

(Remember: To verify an equality you should consider the left-hand and right-hand sides separately.)

Hint.

Start on the left, and apply the fact that \(\varphi\) is a Bijective correspondence to the element \(y\) in \(G'\text{.}\)

(b)

Convince yourself that a similar argument would establish

\begin{equation*} y \cdot \varphi(e) = y \text{.} \end{equation*}
(c)

Explain why the equalities established in Task a and Task b imply that we must have

\begin{equation*} \varphi(e) = e' \text{.} \end{equation*}
Hint.

Apply the conclusion of Discovery 2.7 to the group \(G'\) and its identity \(e'\text{.}\)

Discovery 7.6. Isomorphism preserves (positive) powers.

Use the operation-preserving property of \(\varphi\) to argue that the equality

\begin{equation*} \varphi(x^k) = \varphi(x)^k \end{equation*}

should hold for each group element \(x\) in \(G\) and each positive integer \(k\text{.}\)

Remark 7.2.2.

Taking Discovery 7.5 into account, and recalling the convention that \(g^0 = e\) in every group, we can consider the result of Discovery 7.6 to hold for all non-negative integers \(k\text{.}\)

Discovery 7.7. Isomorphism preserves order.

(a)

Suppose group element \(x\) in \(G\) has finite order \(n\text{.}\) Verify that \(\varphi(x)\) also has order \(n\) in \(G'\text{.}\)

Hint.

There is more to verify here than just \(\bbrac{\varphi(x)}^n = e'\) (though you should indeed verify that first).

(b)

Suppose group element \(y\) in \(G'\) has finite order \(n\text{,}\) and suppose \(x\) is the corresponding group element in \(G\) so that \(\varphi(x) = y\text{.}\) Verify that \(x\) also has order \(n\text{.}\)

Hint.

This task is not just a repeat of Task a.

(c)

Argue that if \(\varphi(x) = y\) and one of \(x\) or \(y\) has infinite order then so must the other.

Discovery 7.8. Isomorphism preserves inverses.

Suppose \(y\) is a group element in \(G'\text{,}\) and \(x\) is the corresponding group element in \(G\) so that \(\varphi(x) = y \text{.}\)

(a)

Verify that

\begin{equation*} \varphi(\inv{x}) \cdot y = e' \text{.} \end{equation*}
(b)

Similarly verify that

\begin{equation*} y \cdot \varphi(\inv{x}) = e' \text{.} \end{equation*}

Careful: Do not assume that either of \(G\) or \(G'\) is Abelian.

(c)

Explain why the equalities established in Task a and Task b imply that we must have

\begin{equation*} \inv{y} = \varphi(\inv{x}) \text{.} \end{equation*}
Hint.

Apply the conclusion of Discovery 2.8 to the group element \(y\) in \(G'\text{.}\)

Discovery 7.10. Isomorphism preserves commutativity.

Argue that if one of \(G\) or \(G'\) is an Abelian group, then so is the other.

Subsection Recognizing groups that are not isomorphic

By considering what should be preserved under an isomorphism, as explored in the preceding subsection, we can demonstrate that two groups cannot be isomorphic by coming up with just one example of some property that is not preserved.

Discovery 7.11.

Way back in Discovery 1.11, we looked for ways to identify the differences in the rotational symmetries of the shapes of the tetrahedron, the flat hexagonal plate, and the pyramid with dodecagonal base, by comparing the algebra of composing rotation transformations for each of the shapes. We can now make this idea more precise: argue that no two of the rotational symmetry groups of these three shapes can be isomorphic by identify some group algebra property in each group that does not occur in the other two groups, but which would be preserved if the groups were isomorphic.

Discovery 7.12.

Verify that the additive group \(\R\) cannot be isomorphic to the multiplicative group \(\R^\times\) (where \(\R^\times\) denotes the collection of non-zero real numbers), by identifying some group algebra property possessed by one of the groups but not the other.

Note: Similarly to what we saw in Example 7.1.4, it is not a valid argument to merely say that \(\R^\times\) have “one fewer” element than \(\R\) (namely, \(0\)), and try to conclude that the two collections \(\R^\times\) and \(\R\) cannot be in bijective correspondence because they “don't have the same number of elements.” (What does “one fewer” than infinity mean, anyway?)