# Determine whether these algebraic structures are isomorphic

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

• Wait, those structures are clearly isomorphic - is your question if that particular maps yields is an isomorphism between them? – Prince M Feb 20 '19 at 19:03
• 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. – Prince M Feb 20 '19 at 19:06
• @Prince M to clarify, yes, it is actually to check if the particular map yields an isomorphism between the first and second binary structures – Evan Kim Feb 20 '19 at 19:40

## 1 Answer

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 $$\langle G, + \rangle$$.

• 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 $\mathbb{Z}$ – Evan Kim Feb 20 '19 at 19:43
• @EvanKim, yes, it will not always be this easy :) – Prince M Feb 20 '19 at 19:59
• Please see my edit, I give you an exercise. – Prince M Feb 20 '19 at 20:01
• does $\varphi$ represent a 2nd map function? Would I want to show that $\phi$ and $\varphi$ are isomorphic to $\langle G, + \rangle$? – Evan Kim Feb 20 '19 at 20:34
• Yes, $\varphi$ and $\phi$ are two separate functions. Show that both of them are isomorphisms – Prince M Feb 20 '19 at 21:14