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The title is the whole problem, but I don't know what is that mean and what it want me to do?

To find out the properties of $\Bbb Z_{2}\times \Bbb Z_{2}$?

Can anyone explain this to me?

user26857
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梁楷葳
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    Its the automorphism group, the group of isomorphisms $G \to G$ – basket Nov 06 '16 at 21:19
  • @basket Yes, I know that. But what is determine means? – 梁楷葳 Nov 06 '16 at 21:22
  • Maybe it is the flipper function (a,b) > (b,a) and identity function. – Jacob Wakem Nov 06 '16 at 21:26
  • Given that your real question seems to be, what does the group of automorphisms mean, you might not find the answer previously given to "the title is the whole problem" (simply because you lacked the basic definitions to understand that problem). In the future I urge you to consider what specific point you understand how to ask about, so that learning the answer will advance your understanding. Here it would help to know the group of automorphisms of $G$ is itself a group (quite possibly not the same as $G$ itself). – hardmath Nov 06 '16 at 21:46

1 Answers1

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Note that $\mathbb{Z}_2 \times \mathbb{Z}_2$ can be thought of a vector space, over the field $\mathbb{Z_2}$. Also, group homomorphisms $T : \mathbb{Z}_2 \times \mathbb{Z}_2 \to \mathbb{Z}_2 \times \mathbb{Z}_2$ can be thought of as linear transformations (why?). Since linear transformations can be represented as matrices, we conclude that the ring of homomorphisms of $\mathbb{Z}_2 \times \mathbb{Z}_2$ can be identified with the ring of $2 \times 2$ matrices with entries in $\mathbb{Z}_2$.

I agree that the question is a bit vague, but I think probably what is wanted is that for you to

  1. Realize the maps of $\mathbb{Z}_2 \times \mathbb{Z}_2$ can be described by matrices.
  2. Interpret the automorphism group in terms of this picture.
Mike F
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