1 Cor. 11:10

Here is verse 10 of this chapter in the Greek.

διὰ τοῦτο ὀφείλει ἡ γυνὴ ἐξουσίαν ἔχειν ἐπὶ τῆς κεφαλῆς διὰ τοὺς ἀγγέλους

because of this ought a woman power to have on the head because of the angels/messengers

This is why a woman should have authority over her own head: because of the angels. ISV

It is important to realize what is properly understood as a correct literal translation. First, it does say "power", but this could mean "power" or "a power" as Luther puts it, "ein Macht" and the Dutch has a similar expression. Not that it makes sense, but it could be correct. Second, it would still be perfectly literal to add "on her head". That also is linguistically possible. However, it is not literal to add "a sign of" authority. That is simply not there.

It is perfectly accurate to say that ἐξουσίαν means any of the following, power, authority, freedom, permission, liberty, license.

Here is an interesting use of the word.

ἐπὶ τῇ τῆς εἰρήνης ἐξουσίᾳ

certain Greek citizens, including Aeschines, were availing themselves of the liberty of the peace to visit Macedonia and take bribes Demosthenes

So it becomes "the liberty of the peace". Therefore, why not "liberty of the head". I can't solve this now, but this is a good question to ask. The main point is that when someone says that the translation of certain verses betray a bias, that is perfectly accurate. I cannot tell you what the 'correct' translation is, but if women had been in charge of translating the Bible for the first two millenia, this verse might have read,

This is why a woman should have freedom concerning her own head, the liberty to do as she thinks right about her head, because of the angels.

Once again here is the post from Metacrock on this. Another option he mentions is that "power on the head" actually means to have authority oneself. To be a bearer of authority. Is it possible that being married gives one authority? But that would contradict Paul's recommendation to stay single. Is it simply because of the place that a woman has in God's kingdom that she has authority. Possibly.

In any case, in this passage, the woman is in the assembly and she is praying and prophesying.

Why associate this chapter with subjection and silence when it mentions authority and prophesying? Think of Huldah and Deborah, two women of the Old Testament, a prophetess and a judge. Should not women in the New Testament have at least as much liberty as those in the Old?