Is it even islamic to have friendship between the man and woman?
It's better for Muslims to avoid close friendships with the opposite gender,
Even though people's intentions are pure, attraction between the sexes is natural. It would be very unusual for a group of young (or even older!) people to spend much time together without attractions developing. Flirting is almost impossible to avoid. You might feel that this type of friendship is unlikely to lead you astray, but it might well have an ill effect on others.
It's preferable that people make friends with their own gender and avoid situations that might cause fitnah.
By the way, there are plenty of Muslims who feel that men and women should avoid each other's company unless they are mahrems.