how to compare two groups I need to compare two groups of people, where the independent variable is having a college degree, and the dependent variable is the income.

The problem is that if I divide the whole population of the study into two groups,
one of them has signally more persons, so the mean is affected by the size of each group and the value of the outliers. 

How can I demonstrate or reject the theory that having a college degree 
among the population of the study, ensures a better income?
I assume you are performing an independent samples t-test. That the Ns are different isn't necessarily a problem -- the mean is an unbiased estimator -- but just how different are the sample sizes?
As you describe, you are probably violating the homogeneity of variance assumption. 

T-tests tend to be fairly robust to this violation, but you might consider having some outlier treatment and/or using a nonparametric test.