(Again, I am sorry for the simplicity of these questions. I am a mechanical engineer learning this stuff late in life.) I have read the following terms or phrases: group algebra group algebra the algebra of a group an algebra group an algebraic group a group of algebras So.... could someone please explain this to me? I understand the stand-alone definitions. But this mixing of terms completely baffles me. Could some one please take a moment to give a coherent and stand-alone definition of each of these things... You can go easy on the math -- I am slowly getting it (I think): I have an understanding of groups, rings, fields, etc. I am as interested in the syntax of how these words are used. Are they nouns or adjectives or adverbs? Yes, that is the level of my confusion. Is algebra a "discipline of math" or " a "thing" or a "class of things." Here is an example: I have read: Lie algebras are closely related to Lie Groups (Again, I am sorry: I KNOW these are stupid questions.) I can say a Macintosh and a Golden Delicious are closely related because they are both apples. I can say a computer and a desk are closely related because they are both things. But I cannot say that GREEN and LOVE are related: they are different things. So what is it in the definition of groups and algebras that can be related?