Why Groups Don’t Have Dimensions

To understand why groups don’t have dimensions, consider why vector spaces do have dimensions.

In any vector space, various numbers coincide:

All minimal spanning (generating) sets have the same size.
All maximal (linearly) independent sets have the same size.
All independent spanning sets (bases) have the same size.
All maximal chains of substructure inclusions have the same size.
All four sizes described above are equal.

It is the common size of all the constructs listed above that gives rise to the dimension of a vector space.

Groups do not have that property. Consider various examples.

The integers \(\Bbb Z\) under addition. \(\Bbb Z\) is cyclic, generated by {1} and by {–1}, each of which is clearly independent and minimal. It is also generated by, say, {2, 3}. {2, 3} is also independent and minimal: No proper subset generates \(\Bbb Z\) and neither number is in the “span” of the other. Also generating \(\Bbb Z\) is {6, 10, 15}, which is independent and minimal. ({6, 10} generates 2\(\Bbb Z\) , {6, 15} generates 3\(\Bbb Z\) , and {10, 15} generates 5\(\Bbb Z\) .) Clearly, we can continue the pattern and find minimal, independent, generating subsets of any finite size. Each of these is also a maximal independent set. Lastly, we observe that \(\{0\}\mathbin{\hat{\subseteq}}\cdots\mathbin{\hat{\subseteq}}8{\Bbb Z}\mathbin{\hat{\subseteq}}4{\Bbb Z}\mathbin{\hat{\subseteq}}2{\Bbb Z}\mathbin{\hat{\subseteq}}{\Bbb Z}\), which is a substructure inclusion chain of infinite length. We have no way to assign \(\Bbb Z\) a dimension in a sense analogous to a vector space dimension.

The square’s dihedral group, D8. Write D8 = {E, R1, R2, R3, FH, FV, D1, D2} for the identity, the three nontrivial rotations, and the horizontal, vertical, and diagonal reflections. R1 and any reflection generate D8. Indeed, all independent generating subsets have size 2. However, the longest substructure inclusion chains have length 3, for example, \(\{E\}\mathbin{\hat{\subseteq}}\{E,R2\}\mathbin{\hat{\subseteq}}\{E,R1,R2,R3\}\mathbin{\hat{\subseteq}}D8\). (We count the inclusions, not the structures, hence length 3.) As with \(\Bbb Z\) above, there is an ambiguity in what number to assign as a dimension for D8.

The best we can do is the smallest size of any generating set, which is unsatisfying.