Question

Choose the WRONG statement

• A. Every subgroup of an abelian group is normal
• B. In a factor group $G/H$, the subgroup H is not normal
• C. A subgroup of index 2 of a group is normal
• D. A subgroup H of a group G such that $aH = Ha, \forall a \in G$ is normal

Hint:

Normal subgroups are the "gatekeepers" of factor groups—you can't have the group $G/H$ without $H$ being normal.

Answer 1

The Correct Option is B

Step 1: Concept
A subgroup $H$ is normal in $G$ if its left and right cosets are equal ($aH = Ha$).

Step 2: Meaning
A factor group (or quotient group) $G/H$ is defined if and only if $H$ is a normal subgroup of $G$.

Step 3: Analysis
Statement (B) claims $H$ is not normal in a factor group. This is a contradiction, as the very existence of $G/H$ requires $H \trianglelefteq G$.

Step 4: Conclusion
Statement (B) is mathematically incorrect and is therefore the "wrong" statement requested. Final Answer: (B)