Underarm odor is a natural process that occurs when bacteria found on the surface on of skin naturally breaks down the sweat that we emit and is expected when someone reaches puberty usually around their first decade of life. When odors are smelt in children much younger than a typical age of puberty, there are two main directions that the investigation can go down: hormonal or non-hormonal.
An unfortunate truth is that it is entirely possible for young children to experience puberty at an age much sooner than expected. Two reasons that this is possible to happen are due to potential endocrine disorders in young patients or because of obesity. Children can be born with sex organs that are functioning much earlier than expected due to changes in their DNA. Puberty is more likely to occur at a younger age in Hispanic and African-American populations compared to Caucasians in North America without a definite medical explanation known.
This term is coined "precocious puberty" and it is not as rare as it used to be. Current epidemiological studies suspect that breasts will begin to form in girls less than 8 years old in 3 percent of cases and testicles will begin to enlarge in 4 percent of boys before the age of 9.  It is best if parents meet with a specialist in order to determine if this is likely the cause in these children.
Another factor that can lead to precocious puberty in young patients can be rooted in the diets that they eat. Western diets high in sugars and fats can predispose children to having high levels of adipose tissue in their body. Unfortunately, this is not the innocent "baby fat" that children can easily grow out of and high levels of this fat tissue can cause the early production of estrogen in patients, even males. Estrogen will cause puberty to occur at an earlier age and will most definitely cause body odor to occur at a younger age. 
An alarming trend in the population in the rapidly climbing incidence of childhood obesity and type 2 diabetes. Parents need to do much better jobs at determining how much sugar is in the diets that their children are eating and should not be tricked into buying foods labeled as "low fat" which usually have much higher levels of sugar to add flavor. This will make a significant improvement in the reduction of preventable precocious puberty cases and let children have a higher quality of lives.
The last avenue worth mentioning could be seen in children who have no obvious health concerns. If your child is not overweight or if they are not undergoing obvious signs of puberty, superficial skin infections can be the culprit for underarm odor. If children do not frequently bathe, the skin under the arms can begin to grow foul-smelling bacteria. Making sure that a child cleans themselves regularly and does not wear clothes for multiple days are two easy ways to prevent this from occurring. Consult with your dermatologist as well to make sure topical antibiotics can be applied if necessary.
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