Question

[UNKNOWN] Given a group G with infinite order, is it fast to find the inverse of any given element in G?

This question arose in one of my math classes today and no one knew the answer. Obviously, if G can also be written as a field, we can use the Extended Euclidean Algorithm. But what if the group can't be written as a field? Is there still a fast way to find the inverse?

Answer 1

good question, i have no answer though lol

Answer 2

@Zarkon @satellite73 @FoolForMath Have any ideas?

Answer 3

@UnkleRhaukus @imranmeah91 @myininaya @amistre64 @saifoo.khan @Mr.Math

Any ideas of if this true or not? And/or do you know who on here would be most likely to be able to help?

Answer 4

dr kiss would know im sure, this is sounds like her type of abstract algebra thing

Answer 5

I'm not sure I understand your question right, I don't know what exactly you mean by "fast way". This seems to be relative for how complex or simple is the group. For instance, it's very easy to find the inverse of a cyclic groups. Take for example the set of real numbers under addition or the set of positive rational numbers under multiplication. I hope I'm not off topic.

Answer 6

@across might be interested in this.

Answer 7

*to find the inverse of an element in a cyclic group*

Answer 8

I'm using "fast" as in polynomial time. For any finite group this is fast using successive squaring and Lagrange's Theorem, but if you're in an infinite group, we didn't know of any algorithm to find the inverse of an element.

In some infinite groups, like the rationals or the integers, it is easy to find the inverse since they're cyclic, or for a variety of other reason, but in others, is it still easy?