The correct option is E Allolactose
Repressors are the transcription factors that bind to operator gene and prevent the RNA polymerase from binding the downstream promoter genes and thus prevent their transcription. Inducer is a molecule that binds to repressor and cause conformational changes in repressor which in turn can no longer bind to operator. In presence of lactose, allolactose; an isomer of lactose, binds to the repressor and causes an allosteric shift. The altered repressor cannot bind to the operator, allowing RNA polymerase to transcribe the lac genes. Thus, allolactose serve as inducer of lac operon. Correct answer is D.