## Review of Group Theory: Group Actions (Pt. III G-Space Isomorphisms and the Fundamental Theorem of G-Spaces)

**Point of post: **In this post we describe what it means for two -spaces to be *isomorphic *and describe the *Fundamental Theorem of G-Spaces.*

*Motivation*

Just as with all other structures it’s fruitful to define the maps between -spaces that preserve the structure we’re interested in. It’s intuitively clear that for -spaces this kind of map should preserve the action, in the sense that if we act on an element and then map it over we should get the same result if we map it over and then act on it.

After we have rigorously defined the notion of -space isomorphism we describe the *Fundamental Theorem of G-spaces*.

**-space Isomorphisms**

Let and be two -spaces with -actions and respectively. We say that and are *-space isomorphic *if there exists a bijective map such that

for all and . We call such a a *-space *isomorphism.

Some theorems which become immediately obvious are

**Theorem: ***Let and be -spaces with -actions and respectively. Then, if is a -space isomorphism then*

*for every . In particular *

**Proof: **We merely note that

The fact that then follows immediately from the fact that is a bijection.

**Corollary: ***Let and be -spaces and a -space isomorphism. Then, for every it’s true that*

*Transitive -actions and the Fundamental Theorem of G-spaces*

Let be a -space with -action . We call a *transitive -space *if for every . The first theorem which results from considering transitive -spaces is

**Theorem: ***Let be a group and . Then, the -action turns into a transitive -space.*

**Proof: **We merely note that by definition that for any

since is a bijection (this is Cayley’s Theorem). The conclusion follows.

and with this we are able to state the *Fundamental Theorem of -spaces*

**Theorem: ***Let be a transitive -space with -action . Then, for any acted upon by the -action is -space isomorphic to .*

**Proof: **For a fixed defined to be

Then, consider that if then if and only if in other words . Thus, it follows that and thus the set of are precisely the elements of . Thus, consider the map

we have then that

since it’s evident that is a bijection the conclusion follows.

**Corollary: ***Let be a transitive finite -space with , and . Then, . In particular .*

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

3. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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