# Let g be an element of a group G. If |G| is finite and even, show that g neq 1 in G exists such that g2 = 1

Question
Analyzing categorical data

Let g be an element of a group G. If $$|G|$$ is finite and even, show that $$g \neq 1$$ in G exists such that $$g^2 = 1$$

2021-03-03

Pair up if possible each element of   G with its inverse, and observe that
$$g^{2}$$ not equal $$e\Leftrightarrow$$ not equal $$g^{−1} \Leftrightarrow$$ there exists the pair $$(g,g^{−1})$$
Now, there is one element that has no pairing: the identity   e (since indeed   $$e=e−1 \Leftrightarrow e2=e,e=e −1$$ $$\Leftrightarrow e^{2} =e$$), so since the number of elements of   G is even there must be at least one element more, say   e not equal a\in G  e not equal a\in G, without a pairing, and thus   $$a=a−1 \Leftrightarrow a2=e.a=a^{−1} \Leftrightarrow a^{2} =e$$.

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Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
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The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
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(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
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At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
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Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
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lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.