-1

Find an infinite group in which every element $g$ not equal to $e$ has order $2$. In this case $e$ is the identity element.

I have tried so many times but i can't really seem to understand what the question wants. Can someone please help me out in solving this question.

Seirios
  • 33,157
Shazia
  • 1
  • 1
    Hint: this implies this space is an $\mathbb{F}_2$ vector space. Can you build an infinite size one? – Alex Youcis Oct 27 '13 at 08:52
  • 1
    Consider $(\mathcal P(S), \Delta)$ for a suitable set $S$ – Hagen von Eitzen Oct 27 '13 at 09:03
  • I can not seem to understand. It would be very helpful if you could actually show me by doing what you actually mean. – Shazia Oct 27 '13 at 09:11
  • Do you know what a group is? Do you know what an infinite group is? Do you know what the order of an element is? Do you know any finite groups in which every non-identity element has order 2? I think we have to find out what you do understand before we can help you with what you don't understand. – Gerry Myerson Oct 27 '13 at 09:38
  • @gerry I do understand those concepts but i can not really seem to be able to solve this question. Can you give me a realistic example which i can use to solve this out please. It would be much helpful if yopu could actually show me how to solve it :) – Shazia Oct 27 '13 at 09:44
  • Alex and Hagen have already shown you how to solve it, but you don't understand their solutions, which is why I'm trying to get to what you do understand. So; do you know any finite groups in which every non-identity element has order 2? – Gerry Myerson Oct 27 '13 at 09:49
  • @gerry i actually have done the solution and i seem to know what they are explaining me. the problem is that i am facing some difficulties in writing it over here in mathematical form which is a reason why i am asking if anyone can solve it directly so that i could verify my solution with you. As for your question then i know that i have to look for a finite group maybe H and then find the infinitely many copies of it which i have done. Is there any way you could solve the questions and i am able to check it with my solution. – Shazia Oct 27 '13 at 09:58
  • Why don't you provide your attempt to solve it? – ypercubeᵀᴹ Oct 27 '13 at 10:03
  • Oh god just leave it. I better submit what i have done so far. I am not submitting because i have difficulties in typing it here as it is very hard for me to type here as i am a new user. – Shazia Oct 27 '13 at 10:10
  • Too late now that the question has been closed, but how would seeing my solution help you to verify yours, if mine turns out to be completely different from yours? For future reference, the best way to verify your solution to a problem is to present your solution and see what people tell you about it. If you're having difficulty with formatting mathematics on this website, there's a "help" link you can click on, and I think it takes you to other links with information on how to format math here. Hope to see you back here soon. – Gerry Myerson Oct 27 '13 at 11:47

1 Answers1

2

For any index set $I$, all non-identity elements of the direct product $$ G = \bigoplus_{i\in I} \mathbb{Z}/2\mathbb{Z} $$ have order $2$. If $I$ is infinite, then $G$ is infinite, too.

azimut
  • 22,696