this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research and development.

May we ask a conceptual question about conjugacy in algebra. In group theory, two elements $a$ and $b$ of a group are conjugate if there is an element $g$ in the group such that $b = g^{–1}ag$. This gives an equivalence relation whose equivalence classes are called conjugacy classes.

We read in the Wikipedia article that "members of the same conjugacy class cannot be distinguished by using only the group structure." What does that mean in a rigorous sense?

As an additional question, as the conjugacy classes define equivalence classes, they should generate a normal subgroup. What is the conceptual interpretation of this subgroup? Does it allow a geometric interpretation?

Thank you very much in advance.

(we included edits in our question based on the comments; thank you for your comments, for which we are very grateful)

this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research.

May we ask a conceptual question about conjugacy in algebra. In group theory, two elements $a$ and $b$ of a group are conjugate if there is an element $g$ in the group such that $b = g^{–1}ag$. This gives an equivalence relation whose equivalence classes are called conjugacy classes.

We read in the Wikipedia article that "members of the same conjugacy class cannot be distinguished by using only the group structure." What does that mean in a rigorous sense?

As an additional question, as the conjugacy classes define equivalence classes, they should generate a normal subgroup. What is the conceptual interpretation of this subgroup? Does it allow a geometric interpretation?

Thank you very much in advance.

(we included edits in our question based on the comments; thank you for your comments, for which we are very grateful)

this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research and development.

May we ask a conceptual question about conjugacy in algebra. In group theory, two elements $a$ and $b$ of a group are conjugate if there is an element $g$ in the group such that $b = g^{–1}ag$. This gives an equivalence relation whose equivalence classes are called conjugacy classes.

Question: We read that "members of the same conjugacy class cannot be distinguished by using only the group structure." What does that mean?

As an additional question, as the conjugacy classes define equivalence classes, they should define a normal subgroup. If this is correct, what is the conceptual interpretation of this subgroup? Does it allow a geometric interpretation?

Thank you very much in advance.

this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research and development.

May we ask a conceptual question about conjugacy in algebra. In group theory, two elements $a$ and $b$ of a group are conjugate if there is an element $g$ in the group such that $b = g^{–1}ag$. This gives an equivalence relation whose equivalence classes are called conjugacy classes.

We read in the Wikipedia article that "members of the same conjugacy class cannot be distinguished by using only the group structure." What does that mean in a rigorous sense?

As an additional question, as the conjugacy classes define equivalence classes, they should generate a normal subgroup. What is the conceptual interpretation of this subgroup? Does it allow a geometric interpretation?

Thank you very much in advance.

(we included edits in our question based on the comments; thank you for your comments, for which we are very grateful)