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Show that $\phi$ is an isomorphism of $\langle \mathbb{Z}, + \rangle$ with $\langle\mathbb{Z}, +\rangle$, where $\phi(n) = -n ~$for$ ~n \in \mathbb{Z}.$

I know that to show that these two binary structures are isomorphisms, I have to show that $\phi(n) = -n$ is one-to-one and onto. I have experience finding isomorphisms of matrices in linear algebra, where I used dependent/independent matrix rows to determine whether the matrices were invertible and thus isomorphic.

For this problem, I am unsure how to just get started, mainly showing how these binary structures are one-to-one and onto.


exercise: Let $\langle G, + \rangle$ be any 'binary structure' (group?). Show that $\phi(g) = -g$ and $\varphi(g) = g$ are both always automorphisms on $\langle G, + \rangle$.

Let $g, g_1, g_2 \in G$ and $\phi(g) = -g$. This is kind of like an inverse property for G.
- $\phi(g)$ is onto because we can take the inverse, $\phi(-g) = g$ and $g \in G$.
- $\phi(g)$ is one-to-one because $\phi(g_1) = \phi(g_2) \rightarrow g_1 = g_2$. The $\phi$ function produces the same result for $g_1 \textrm{ and } g_2 \in G$.
- $\phi(g_1 + g_2) = -(g_1 + g_2) = (-g_1) + (-g_2) = \phi(g_1) + \phi(g_2)$ Binary structures are homomorphic(?) to each other

Let $g, g_1, g_2 \in G$ and $\varphi(g) = g$. This is essentially the identity property for G.
- $\phi(g) = g \textrm{ and } \phi(g) = g$ (switch around the input/output but notation isn't clear) for onto check.
- $\phi(g_1) = \phi(g_2) \rightarrow g_1 = g_2$ (passes one-to-one check)
- $\phi(g_1 + g_2) = (g_1 + g_2) = (g_1) + (g_2) = \phi(g_1) + \phi(g_2)$ (homomorphism check)

Comments - That felt straight forward, but it felt more like I was adapting the mechanics of what you had done rather than strengthening the conceptual aspect

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    $\begingroup$ Wait, those structures are clearly isomorphic - is your question if that particular maps yields is an isomorphism between them? $\endgroup$ – Prince M Feb 20 '19 at 19:03
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    $\begingroup$ The structures aren't `isomorphisms', the structures are 'binary structures' (I don't know what this is, are you viewing them as monoids? groups?) and being 'isomorphic' is a binary relationship between two structures (because they either are, or are not, isomorphic). Lastly, in addition to checking one-to-one and onto you also likely want to check some type of structure preservation of the function $\phi$, which will depend on the algebraic structure you are viewing $\langle \mathbb{Z}, + \rangle$ as. $\endgroup$ – Prince M Feb 20 '19 at 19:06
  • $\begingroup$ @Prince M to clarify, yes, it is actually to check if the particular map yields an isomorphism between the first and second binary structures $\endgroup$ – Evan Kim Feb 20 '19 at 19:40
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I think you are viewing $\langle \mathbb{Z}, + \rangle$ as an additive group, and I think your question is to check that $\phi(n) = -n$ yields an isomorphism from $\langle \mathbb{Z}, + \rangle$ to $\langle \mathbb{Z}, + \rangle$. If not, let me know and I will edit my answer. If so, indeed we need to check that $\phi$ is one-to-one and onto, as you mentioned, and also need to check that it is a homomorphism which would mean that $\phi(n_1 + n_2) = \phi(n_1) + \phi(n_2)$.

one-to-one: Suppose $\phi(n_1) = \phi(n_2)$, then $-n_1 = -n_2$ which implies $n_1 = n_2$, thus by definition $\phi$ is one-to-one.

onto: Let $n$ be any element of $\mathbb{Z}$. See that $-n$ is also an element of $\mathbb{Z}$ and $\phi(-n) = -(-n) = n$ so that $\phi$ is onto.

homomorphism: $\phi(n_1 + n_2) = -(n_1 + n_2) = (-n_1) + (-n_2) = \phi(n_1) + \phi(n_2)$.

So $\phi$ is indeed an isomorphism.


remark: When an isomorphism starts and ends on the same structure, like $\phi \colon A \to A$, we usually call $\phi$ an automorphism. Automorphism is just a special type of isomorphism.


exercise: Let $\langle G, + \rangle$ be any 'binary structure' (group?). Show that $\phi(g) = -g$ and $\varphi(g) = g$ are both always automorphisms on $\langle G, + \rangle$.

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  • $\begingroup$ okay, that seems clear enough, thanks for the clarifications. It seems almost too simple to show a one-to-one and onto relationship, but maybe that is just because both binary structures were both in $\mathbb{Z}$ $\endgroup$ – Evan Kim Feb 20 '19 at 19:43
  • $\begingroup$ @EvanKim, yes, it will not always be this easy :) $\endgroup$ – Prince M Feb 20 '19 at 19:59
  • $\begingroup$ Please see my edit, I give you an exercise. $\endgroup$ – Prince M Feb 20 '19 at 20:01
  • $\begingroup$ does $\varphi$ represent a 2nd map function? Would I want to show that $\phi$ and $\varphi$ are isomorphic to $\langle G, + \rangle$? $\endgroup$ – Evan Kim Feb 20 '19 at 20:34
  • $\begingroup$ Yes, $\varphi$ and $\phi$ are two separate functions. Show that both of them are isomorphisms $\endgroup$ – Prince M Feb 20 '19 at 21:14

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