Bo, that was not my only argument, though I still see the similarity between death penalty and homicide, i. e., both involve death by violence].
It's clearly a different level of violence. Injecting a chemical that sedates one out is rather different than the way their victims often die. I'm not sure I'd really count that as violent, but death takes some effort to achieve. Plus the perp gets to prepare for his death, put affairs in order, often has family members or religious counsel nearby. Yes, it's death but they do try to make it relatively gentle.
The other arguments were
Because killing another human (like torturing any other living thing) is always wrong.
I could advocate that racial discrimination is always wrong. But used to remedy past discrimination (ie affirmative action) it can be a positive. That's the context with the death penalty, as a remedy/punishment against past (extreme) bad behavior.
And beyond torturing, I believe simply any intentional killing of animals is wrong.
I'm a pretty hard core vegetarian. Even cockroaches and other bugs I try to catch and let outside (actually I found the trick is to find the cockroach egg sacs, which I put outside). I do kill mosquitoes, who are after my very own lifeblood and fuck with my sleep (I'm allergic to their anti-coagulants).
Justice for the victim and their family? How about revenge in her name? She's not coming back. best we can do is make sure the perpetrator can't do it again.
Death penalty and eye-for-an-eye justice has a long tradition throughout human societies. Maybe it's partly the simple symmetry which appeals to notions of fairness.
Further, families of murder victims frequently say that the execution of the killer provides a sense of closure and justice. Also, when family members are against the death penalty, that should be taken into consideration.
It's too easy to focus on the killer and his rights since he is alive and the victim is long gone. But there is a fundamental unfairness between the killer who gets to live another 20 - 50 years of his life, albeit in prison, sharing family milestones and voting Democratic and eating food and reading books and living. While their victim, had their life and future snuffed out. Really that unfairness seems to me one of the strongest arguments in favor of capital punishment. And you can build in an automatic 10 year period from death sentence to execution, to allow for appeals, new evidence, etc -- which is already the de facto system. The killer getting 10 years more of life than their victim isn't at all fair, but provides some balance and a chance to correct any possible injustice.
Deterrence? According to most research, it doesn't work that way. Many murderers prefer death to life behind bars.
Suicide-bent killers are a big problem. Usually folks care what happens to themselves and won't put their lives in jeopardy. Murderers are often young people, not that well educated, have violent or neglectful backgrounds and don't think much of the consequences. Ie hard to deter. In those/most cases deterrence won't work. Prison in America is such a violent and unpleasant experience, life in jail doesn't sound much better than death. And still our violent max security prisons don't seem to provide much deterrence. So deterrence likely remains mostly theoretical.