Some researchers, like Dr. Arthur Beaudet, a professor at the Baylor College of Medicine and the chairman of its department of molecular and human genetics, applaud the effort. He believes that the acts committed by men like Mr. Lanza and the gunmen in other rampages in recent years — at Columbine High School and in Aurora, Colo., in Norway, in Tucson and at Virginia Tech — are so far off the charts of normal behavior that there must be genetic changes driving them.

“We can’t afford not to do this research,” Dr. Beaudet said.

Other scientists are not so sure. They worry that this research could eventually stigmatize people who have never committed a crime but who turn out to have a genetic aberration also found in a mass murder.

Everything known about mental illness, these skeptics say, argues that there are likely to be hundreds of genes involved in extreme violent behavior, not to mention a variety of environmental influences, and that all of these factors can interact in complex and unpredictable ways.

“It is almost inconceivable that there is a common genetic factor” to be found in mass murders, said Dr. Robert C. Green, a geneticist and neurologist at Harvard Medical School. “I think it says more about us that we wish there was something like this. We wish there was an explanation.”

Scientists are well aware of the fraught history behind the questions of biology and violence.

In the early 20th century, claims that criminal behavior was inherited arose during the eugenics movement and led to sterilizations of mental patients and felons.