If O(G) = 21 and G is non abelian then the total no. of elements in centre of G is ________.Correct answer is '1'. Can you explain this answer?

Question

Answer

if O(G) = 21 and G is non abelian then by a well known result we know that

which implies G is abelian but given that G is non abelian.

Hence only possibility is 0[z(G)] = 1 i.e z(G) = {0}

Answer

Explanation:

Given Information:
- O(G) = 21
- G is non-abelian

Center of a Group:
- The center of a group G, denoted by Z(G), is the set of all elements that commute with every element of G.
- In other words, Z(G) = {g in G | gx = xg for all x in G}.

Non-Abelian Group:
- A group is non-abelian if there exist elements a, b in the group such that ab is not equal to ba.

Order of the Center:
- For any group G, the order of the center Z(G) divides the order of the group, i.e., |Z(G)| divides |G|.
- Also, if G is non-abelian, then Z(G) is a proper subgroup of G.

Given O(G) = 21:
- Since O(G) = 21, the order of the group G is 21.
- Since G is non-abelian, the center Z(G) is a proper subgroup of G.

Conclusion:
- The only proper subgroup of a group of order 21 is the trivial subgroup {e}, where e is the identity element.
- Therefore, the total number of elements in the center of G is 1, which is the identity element e.

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