Question
I see some answers contain suggestions of replanting arms or cuttings from Argentine cactus, but there is little detail.  The "pups" that grow on the cactus would appear to be able to be separated from the main arm, but I don't see that as a method of propagation.  Rather the information is about a "cutting". Please advise of the various methods of "replanting" from an existing cactus.  Thanks, Ron

Answer
I am assuming  that you have one of the Stetsonias which are quite large and branching and grown outdoors?  Part of my response is based on that assumption.  Arms of cactus can be twisted off. They can be heavy so cutting can be easier than twisting and you need really heavy gloves or a piece of carpet. If they are fibrous that can be difficult, if not go ahead and do that,  Just make sure you don't leave any wounds in the mother cactus.  If the end of the arm is totally dry, you can plant it in a pot in the same soil as the original, with some added gravel and slite.  If it's not totally dry you need to let that callous over. Water about 1-2X a month (barring rain).  Keep it in a semi shaded area (or put shade cloth over it) and leave it in that pot for a year.  They can take a very long time to root.

One cuts an arm off because it is too heavy, spiny, neater or is being separated from an infected plant.  In that case one sterilizes a knife, makes a clean horizontal or slightly angled cut.  I dust mine with either fungicide, charcoal powder or rooting powder just to protect it as the cut end calluses, which can take a couple of weeks.  Again, if it is from an outdoor cactus you can use that soil with added slite and gravel.  Same care.

You can also start from seed at the beginning of the plant's active period.  For cactus this is usually in the Spring.  That usually requires overhead lights, warmth above and below, sterilized soil or perlite or sand.  I know that your cactus can be started from seed, if you can find it (Mesa Gardens has great seeds).

The most important factors as far as I'm concerned are disease prevention for both the mother and the pup, appropriate soil, not overwatering and shelter.