aphids are "soft bodied" insects - the soft part in contrast to bugs with a hard shell that are not affected by soapy water.
the soapy water washes off their waxy coating and the aphids die from dehydration.
I advise some care in selecting your "soap" - you must stick with a natural soap - detergents have lots of 'other stuff' in them and they are positively famous for killing people's whole gardens / flower beds. you cannot count on word of mouth / what worked for somebody else / what worked last year because companies routinely change their formulations based on raw materials cost and different geographic areas may have different formulations.
people holler: but my grandmother always just threw the dishwater on the plants - okay, true. but your grandmother likely was not using the better-detergents-through-chemistry of today.
Safers was mentioned - it's a known good bet.
people talk about Ivory, and Dr. B's peppermint, etc.
you can probably find handmade/'natural' bar soaps in a lot of shops - tallow & lye - nuttin else perhaps but some scents.
you can eliminate aphids with plain water - just spray the plant and blast them off.
the soaps are all "contact" killers anyway - the real issue with controlling them is they are born pregnant and reproduce at an incredible pace.
when I get an infestation - on roses or peas for example - I blast them off with the hose _every comma every_ day for about a week.
for an indoor plant. one may be limited in 'facilities' to accomplish that - a spray bottle of insecticidal soap works with less 'mess'