This soreness and redness of armpit can be due to a boil or a furuncle. A boil, also referred to as a skin abscess, is a localized infection deep in the skin. A boil generally starts as a reddened, tender area. Over time, the area becomes firm and hard. Eventually, the center of the abscess softens and becomes filled with pus.
Apply warm compresses on the boil and keep the area clean. Drainage of the boil is done only when it becomes soft and forms a head. Take some over the counter pain reliever and consult a doctor for examination and antibiotics prescription. If there are recurrent boils, hen please get your blood sugar levels estimated.
I hope it helps. Take care and please do keep me posted in case you have any additional doubts. Kind regards.