Swelling of lymph nodes under the arm is a symptom of breast cancer? are the swollen lymph nodes supposed to be pain-free?

Can be: Usually a breast mass would be felt by the time breast cancer has spread to enlarged lymph nodes that can be felt under the arm. Usually the lymph nodes are painless. Any physical change you are feeling should be evaluated with a doctor and likely a mammogram. Many things including infections and cancers can cause enlarged lymph nodes.

No neccasarily: Lymphatic system absorbs/carries things too big to go into the arteries/veins like bacteria eaten by white cells and digested food. Lymph nodes "taste" the lymph for anything bad like bacteria/viruses/cancers, etc. When it detects a problem, it alerts the immune system and your body reacts. They enlarge in response to the inflammation they create when they react. They shrink when all is well.