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Group Structure, Order of two groups

I have questions about constructing a group structure, how to identify the order of a paired group when they have different orders and method of figuring out the group identity and the inverse of a pair that contained in the paired group.

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If G and H are groups then explain how to equip G x H with a group structure. If G has order k and H has order m then what is the order of G x H? What is the group identity and the inverse of an element (g, h) Є G x H.
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Proof:

G and H are groups, then we can construct the product of the two groups GxH. GxH is defined as ...

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This solution is comprised of a detailed explanation to answer what is the group identity and the inverse of an element (g, h) Є G x H.

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