Suppose there exist a person in the group who has either three friends or one friend.
Now,since every person in the group is either a friend or enemy of the others so having only one friend means he has three enemies.
Now, construct a graph where the persons are represented by nodes and edges are joined between the nodes iff they are friends. Let the graph be named as G.
Assume that the node A has three friends B, C and D
Since no three of them are friends there will not exist any triangle in G.
Since A,B and A,C are friends B,C can't be friends anymore otherwise there would be a triangle in G.
Since A,B and A,D are friends B,D can't be friends anymore otherwise there would be a triangle in G.
And since A,C and A,D are friends C,D can't be friends anymore otherwise there would be a triangle in G.
So, we get that B,C and D are enemies but it is given that no three of the group are enemies.
Hence contradiction so, A can't have three friends.
Since A is chosen arbitrarily so none of the members of the group would have three friends.
Now, assume A has only one friend. Then A will have three enemies.Then the graph G' ( complement of G ) would contain three edges from A. Same reasoning as above (just replace friends with enemies) would show that A can't have three enemies either.
So, no one in that group can have only one friend.
So, every member in that group has exactly two friends.