When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does that mean? If y is an element of xH, what do we know about y? If y is also an element of Hx, what do we know about y?  How are these two “facts” linked together by the “=”?

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 22E: 22. If and are both normal subgroups of , prove that is a normal subgroup of .
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When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does that mean?

If y is an element of xH, what do we know about y?

If y is also an element of Hx, what do we know about y? 

How are these two “facts” linked together by the “=”?

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