Question

Let G be finite set, which is a group under some operation and H be a subgroup and $a \in H$. Considering their order cardinality in increasing order :
A. $O(G)$

B. $O(a)$

C. $O(H)$

D. $Card(2^G)$

Choose the correct answer from the options given below :

• A. B, C, A, D
• B. D, B, C, A
• C. D, B, A, C
• D. B, C, D, A

Hint:

In group theory hierarchy: Element $\leq$ Subgroup $\leq$ Group. Power sets always result in the largest cardinality due to exponential growth.

Answer 1

The Correct Option is A

This question applies Lagrange's Theorem and basic set theory regarding the sizes of groups and power sets. 1. Applying Lagrange's Theorem:
• The order of any element $a$ divides the order of the subgroup $H$ it belongs to. Thus, $O(a) \leq O(H)$.
• The order of a subgroup $H$ must divide the order of the group $G$. Thus, $O(H) \leq O(G)$. 2. Set Cardinality:
• $2^G$ represents the power set of $G$. If the group has $n$ elements, the power set has $2^n$ elements. For any finite set, $n < 2^n$. Therefore, $O(G) < Card(2^G)$. 3. Final Sequence: The increasing order is: B (Element) $\to$ C (Subgroup) $\to$ A (Group) $\to$ D (Power Set).