There are always two kinds of induction in general:

* Assume that the result holds for $n-1$
* Assume the the result holds for integers $<n$

For the former one, we only need to check one base case; for the latter one, we always need to check two base cases.

I’m learning group theory. Induction is a very useful tool that is always employed. We always prove by induction on group order $|G|$ and assume that results hold for groups of order $<|G|$. But I find that we always check only one base, namely $|G|=1$! Maybe I misunderstood something, but I really don’t remember seeing any proof checking more than one base case.

So what is the point? Could you give me some ideas? Any help would be appreciated!

Possible examples:

* [Index of maximal proper subgroup of a solvable group](https://math.stackexchange.com/questions/560924/index-of-maximal-proper-subgroup-of-a-solvable-group/560934#comment7120986_560934)
* <https://math.stackexchange.com/a/2391609/792898>