Display & check-for User Group on public profile
If you know the user will only belong to one group, something like this should do it.

{% set user = craft.users.username(craft.request.getSegment(2)).first() %}

{# make sure we found a user %}
{% if user|length %}
    {% set groups = user.getGroups() %}

    {# make sure they belong to at least one group #}
    {% if groups|length %}
        {# display the name of the first group #}
        {{ groups[0].name }}
    {% endif %}
{% endif %}