identity, plus pk?

We have a table that has an identity field along with 5 other domain
fields. The identity field is not declared as a primary key. The
table has 3.5 million records.

A consultant was hired recently to provide insight. His major
recommendation: modify the table to make the identity field a primary
key (i.e., alter table add constraint...)

Is that sound advice? Is it OK to have a table with identity but no
primary keys? What would be the impact on performance?

.

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