Pretty much everyone in the future society depicted in the story has some sort of artificial handicap, from heavy weights to ugly masks. This prevents anyone from excelling at anything (for example, one actress being picked over another for her looks), and so ensures that everyone is "equal," at least physically and mentally. George, Hazel's husband, has a mental handicap that plays loud noises to distract him and prevent him from thinking about things. While it is likely that Hazel has some physical handicaps:
Hazel had a perfectly average intelligence, which meant she couldn't think about anything except in short bursts.
[...]
There were tears on Hazel's cheeks, but she'd forgotten for the moment what they were about.
(Vonnegut, "Harrison Bergeron," tnellen.com)
This shows how the handicap amendments work; a baseline of intelligence was chosen, likely as an average of the population, and then people were handicapped to...
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lower them to that average. Hazel happens to be right on that average intelligence level. George is lowered to Hazel's average, but Hazel needs no lowering, and she can't be handicapped further, because then she would be inferior to the average.
Why isn't Hazel handicapped in "Harrison Bergeron"?
Hazel does not have any handicaps because she is already completely average. She must be average looking, have average intelligence, and have average strength and speed. Due to all of these average attributes, Hazel is exempt from handicaps. In the society in which "Harrison Bergeron" takes place, there are handicaps imposed on people who demonstrated a higher than average ability in any capacity, such as intelligence, appearance, or athleticism. Since Hazel does not exhibit any qualities that take her out of the average category, she does not need to use any handicaps. On the other hand, her husband George needs to wear a radio transmitter that disrupts his thoughts and weighted bags that are used to counteract his strength and speed.
Why are there no handicaps on Hazel in "Harrison Bergeron"?
There are no handicaps on Hazel because she represents society's lowest common denominator. Everyone in the society needs to be handicapped in order for people like Hazel not to feel inferior.
In literature, characters can be symbols, or stand for something larger. Without Hazel, the reader does not know to what level people would have to be handicapped. The narrator says, "Hazel had a perfectly average intelligence, which meant she couldn't think about anything except in short bursts." Meanwhile, Hazel's husband George's intelligence is "way above normal," so he received mental handicaps in his ears.
In general, the intellectually and athletically elite tend to be the models for social behavior. In this society where "everyone was finally equal," the "perfectly average" are the ones who are the models.
Without Hazel, how would the readers know what people like George and Harrison were being handicapped to be like?
Overall, the story is about how the ruling classes in society want people to be perfectly average like Hazel. They want citizens to stay within their roles and not challenge the control systems they have put in place—handicaps, in this case.