Question

Which of the following statements are correct:
A. Subgroup $H$ of a group $G$ is normal in $G$ if $xHx^{-1} \subseteq H$ $\forall x \text{ in } G$ B. A subgroup of an abelian group is not normal.
C. Centre $Z(G)$ of a group $G$ is normal.
D. Every subgroup of an abelian group is normal.
E. Centre $Z(G)$ of a group $G$ is not normal.
Choose the correct answer from the options given below:

• A. A, B, C Only
• B. B, C, E Only
• C. A, C, D Only
• D. C, D Only

Hint:

Every subgroup of an abelian group is automatically normal! Also, $Z(G)$ is always a normal subgroup for any group $G$.

Answer 1

The Correct Option is C

Step 1: Concept:
This question evaluates the properties of normal subgroups and the centre of a group $Z(G)$.

Step 2: Key Formula or Approach:
1. A subgroup $H \le G$ is normal ($H \trianglelefteq G$) if and only if $x H x^{-1} \subseteq H$ for all $x \in G$.
2. The centre $Z(G) = \{z \in G \mid zg = gz, \forall g \in G\}$ is always normal in $G$.

Step 3: Step-by-step Explanation:

• Statement A:
By standard definition, $H \trianglelefteq G$ if $xHx^{-1} \subseteq H$ for every $x \in G$. Hence, Statement A is correct.

• Statement B:
In an abelian group, elements commute, so $x h x^{-1} = h x x^{-1} = h \in H$ for all $h \in H$ and $x \in G$.
Thus, every subgroup of an abelian group is normal. Hence, Statement B is false.

• Statement C:
For any $z \in Z(G)$ and $x \in G$, we have $x z x^{-1} = z x x^{-1} = z \in Z(G)$.
Hence, $x Z(G) x^{-1} = Z(G)$, which proves $Z(G) \trianglelefteq G$. Statement C is correct.

• Statement D:
As established, commutativity implies $g H g^{-1} = H$ for all subgroups $H$ of an abelian group $G$. Thus, every subgroup of an abelian group is normal. Statement D is correct.

• Statement E:
Since $Z(G)$ is always normal in $G$, stating that it is not normal is false. Statement E is incorrect.

Step 4: Final Answer:
Statements A, C, and D are correct. Therefore, option (C) is the correct answer.