TL;DR In the Odyssey, Odysseus son Telemachus would have been 20 when Odysseus returned to find a group of suitors squatting in his home, hoping to marry his wife and inherit his kingdom. But why did the grown Telemachus not take his rightful place as King of Ithaca and banish the squatters once it was decided that his father must be dead?
I hope this is an appropriate question, even though it's technically about a fictional work. But I'm hoping that the answer will shed some light on Greek society at the time. When Odysseus left home for the Trojan War, his son Telemachus was an infant. But Odysseus was away for 20 years and returned to find his home in disarray and filled with potential suitors hoping to marry his wife and (from my understanding) inherit or take control of Ithaca.
But that never made sense to me. Yes, it had been 20 years and The King of Ithaca was assumed dead. And yes, his wife was still alive and probably considered beautiful. But he had an heir! Telemachus by this point would have been at least 20 or 21, and very possibly only a little younger than Odysseus might have been when he originally left (Bonus is anyone knows how old Odysseus actually was when he left home. Early or late 20s? Maybe even 30?) Even in our society that is considered fully grown, and would I be correct in assuming that Telemachus probably would have been considered "of age" for 4 or 5 years at that point? So my question is: why didn't Telemachus take leadership of Ithaca and drive the squatters out of his home? Or at least make it clear that IF his mother were to remarry, Ithaca and it's wealth would still remain his?
Was he simply weak? Or were there some factors within the contemporary Greek Society that prevented him from doing so? Am I misunderstanding the meaning of Odysseus position as King/Leader of his island? Or was it a plot hole in the original story necessary for the narrative to unfold as it did?
And forgive me if I've got any of my premise incorrect. It's been years since I read or watched any adaptation of the Odyssey, but that one question has always bothered me.
The first 4 books of the Odyssey are called the Telemachy. The story of Telemachus is recounted here. It's an introduction to the rest of the Odyssey as it shows what is going on in Ithaca. Telemachus is 21 years of age at this point. He knows he's the heir, but he has very little power. The nobles of the island are the suitors, and in the absence of Odysseus, they have the political power. Telemachus can't do much, because he doesn't have the support he needs to become king.
It should be noted that Laertis, the father of Odysseus is also alive, and he was the previous king of Ithaca (as we learn in the separate tale of the Argonauts). So he either retired and gave up the throne to Odysseus because he was old, or he was retired.
Indeed, it seems that Ithaca's king is proclaimed and monitored by a council. The Council/Assembly of Elders. Telemachus addresses them in book 2 and tries to convince them to remove the suitors from the palace, as they are wasting his inheritance. Presumably, the decision that the next king should be one of the suitors and that Penelope should choose him came from the Council. The suitors also appear to belong to the same families as the Elders.
During the Council, Telemachus opens his statements by admitting his weakness and inexperience. He's still not considered an adult, or at least in a position to be making decisions as a king. He is only a youth. In book 1, his mother is surprised when he orders her to return to her quarters. It's the first sign he is turning into an adult. The Council don't really see that, and they have another political game going, where they want their sons to become kings. So they deny his claim, although some admit to being impressed by him and his arguments. So Telemachus decides he will go on a journey away from Ithaca. This is some sort of rite of passage. By traveling he will gain valuable life experiences and he will mature as a person. When he returns the Council will have to view him differently, as at the very least he will have proven to be able to steer his life without his mother.
In Ancient Greece, adulthood for boys came about generally at 17 or 18. That's not an established age. You were considered an adult if you were deemed old enough to join the military. Something that could happen in a lower or higher age. Telemachus doesn't seem to be part of the military. That's why he's leaving to travel. It'll be his method of becoming an adult. Homer also practices poetic liberty here, in that the events he describes might not really match up to reality, but what's important is that they provide drama (commonly referred to today as the Rule of Cool).
It was common for boys to have a tutor that would teach them what they'd need to become adults. Primarily how to wield a spear and bow, in war and in hunting, as well as how to adhere to a code of honor. Telemachus has no such figure to guide him. Another reason he's yet to become an adult. During his travels he is joined by the goddess Athena (who also favors Odysseus), disguised as an old man who is willing to mentor him. He visits Nestor and Menelaus, who had fought in the Trojan War with his father, and who impart valuable lessons to the boy. And we see him changing throughout the journey. He becomes more self-confident and decisive. It is made clear to him (through the tale of Orestis being compared to his situation) that he needs to take matters into his own hands, not only because it's the best way to be rid of the suitors, but it is also the best way to prove he is a man and worthy to be a leader.
In Book 22, when Odysseus returns, Telemachus is no longer the youth who believed his only course of action was to ask the Council for their help. He stands bravely with his father, firm and resolute, against all the suitors. They are outnumbered, yet he does not waver. Odysseus notes all this and he grows to admire Telemachus and be proud of him. This is when Telemachus becomes a man. His father recognizes him as a full man. He is participating in a battle, in which he is also outnumbered (military service) and he has matured as a thinking and responsible adult. He is ready to take charge of his house and of Ithaca.
I've mentioned the Council/Assembly of Elders. Democracy in Athens didn't develop in a vacuum. All city-states had some sort of Council that ruled alongside whoever counted as king or leader. For example, Sparta had an Assembly that could overrule its two kings. Athens had the Areios Pagos and the Archon Assembly before it instituted democracy. Corinth had their Council. We hear the word "king" and we think something akin to feudal western societies or of Ancient Egypt, but that's not how things worked in Ancient Greece at all.
I know that in Ancient Athens it was highly unusual for men to participate in politics before the age of 30 (G.R. Bugh, 'The Horsemen of Athens', but many other places as well). Of course, different time and place, but it at least makes likely the idea that he wasn't yet considered to be of adult age. I hope someone can give you an actual answer!
From what I can gather, even if he had 20 years, he is considered of an age in which you were not an adult neither a child. Another important factor is that Penelope is still queen, and the position of King can only be solidified if such a tie can be put back together. Telemachus reign hangs on the return of his father, henceforth the importance of the tree of which the bed of Ulysses and Penelope's bed is made.
Source: Jean-Pierre Vernant's The Universe, the Gods, the Men. (This is a direct translation from Spanish which is the version I hold, maybe the title is different in English) I recommend this book if you wish to have an overview of the Greek mythology.