In the book The Chrysalids by John Whyndham, the Sealand woman is superior but how can we prove her arrogance?
'Let him be,' came the severe, clear pattern from the Sealand woman. ... 'Neither his kind, nor his kind of thinking will survive long. ...[T]hey have nowhere more to go. ... The living form defies evolution at its peril; if it does not adapt, it will be broken. ... The Old People brought down Tribulation, and were broken into fragments by it. ... [S]oon they will attain the stability they strive for, ... a place among the fossils.'
A good place to look for the proof of arrogance is to the Sealand woman's speech to David as he debates how he should react to his father, a question he raises in his own mind--and one the Sealand woman answers in his mind--as he sits with Rosalind, Petra and Sophie in the cave that they took shelter in.
Excerpts quoted above, this speech shows her severe judgement against the people to whom David's family belong: the Old People. Her tone is harsh. Her sentences are clipped in anger and superior in righteousness. Her words are terse and without mercy. Her actions in the battle with the Old People follow suit. But what is arrogance and is the Sealand woman truly guilty of arrogance?
Arrogance is derived from an Old French word meaning to have the quality of being overbearing and insolent, of having an exaggerated opinion of oneself, of having an exaggerated idea of one's self-importance.
The defining factor of arrogance, then, is that arrogance is an exaggerated self-assessment, an unrealistic opinion of one's importance and an insolent idea of one's place in the world.
Based on this defining factor, there are those who would argue against a claim that the Sealand woman is arrogant. They would argue instead that she is a powerful woman who rightly knows the strength and importance of her power. She is, after all, heading a rescue mission from far away for a handful of kids in the heart of a wasteland. Her real-world authority and power must be considerable to be able to do this. In addition her telepathic powers must be almost without equal to be able to communicate with, first, Petra, but then with all of them across the expanses.
If she must be identified as arrogant, then the passage represented above will begin to prove this characteristic. Yet it is questionable as to whether she should be evaluated as being arrogant, with an inflated false sense of self-importance, or whether she should be evaluated as truly powerful and actually important. The latter evaluation of her real-world power (as opposed to arrogance) may put more of a chill on the direction evolution will take David and his friends in their new future, a future that may be as cold as their past was hard.