The expression Alpha is quite outdated because our society is rather complex. So it should be taken with some reserve, eg it is difficult for one person to be alpha in more groups. Say you work hard and you become a supervisor at work; that's your alpha position. You still have another bosses above you though, so you are still a beta from their point of view. However, you may go to fitness and compete against heavy lifters. You will be easily beta. You may go to many different groups and have different "position" in each group. Note that it is different among eg wolfs, they simply have only one group and one Alpha....
Nevertheless, Alphas usually have strong opinions and they express themselves differently, so many times they may not like each other (especially if they don't agree on fundamental issues).
A good way is to watch political debates, see the most recent republican one. Each candidate is "Alpha" in his own group, but they have to compete against others on the podium. Donald Trump is currently at Alpha position as he is the most popular, the others rather appear like just barking dogs, most are viciously attacking him everywhere they can to de-throne him and get his position. So who wins? Usually the strongest one, but as you can see it is still quite complex - they no longer compete physically as wolfs do, they have to have good communication skills to express themselves, they have to have good social skills to be likable, they have to be able to form strategic coalitions to win different demographic, they have to have lots of money, they have to be intelligent to push the right buttons at the right time and so on....
In normal life it is not as extreme, but there might still be quite competition. A good Alpha will not try to win by too much dominating the group, he will rather try to form good coalition, lead by example, and make people follow him. Bad Alpha will be more of an asshole, lots of guys will rather avoid him as they don't want to keep fighting constantly for leading position.
Anyway, Alpha position usually requires lots of effort, it gives you good access to girls, but at the same time you are constantly doing extra work. There are better ways to get access to girls, without participating in the whole Alpha position or social ladders....